This document contains additional material for the preprint:
K. Parattu, A. Wingerter, “Tribimaximal Mixing From Small Groups”, arXiv:1012.2842 In Tab. 1 below, we list the 1048 groups of order ≤ 100. The first column gives the GAP IDwhich is a label that uniquely identifies the group in GAP. The first number in the square brackets is the order of the group, and the second number simply enumerates different groups of the same order.
The second column gives the name of the group. If two or more groups by the same name are isomorphic, we list only one. For the conventions we used in naming the groups and for a non- exhaustive compilation of alternative names common in the physics and mathematics literature see Appendix A of arXiv:1012.2842.
The third column indicates whether the group G has a 3-dimensional irreducible representation.
The fourth column indicates whether the group G is a subgroup of U(3). If G is in SU(3), an orange check mark is shown (4), otherwise a blue one (4).
The fifth and sixth columns indicate whether G is in U(2) or U(2) × U(1), respectively (blue check mark). If G is in SU(2) or SU(2) × U(1), respectively, the check mark is orange.
The seventh column indicates whether G contains A4 as a subgroup. The color of the check marks has no significance.
Note that all Abelian groups are trivially contained in some Zp× Zq. . . × Zr, and the dashes (r) indicate that we do not state whether they are subgroups of any U(n).
Table 1: The 1048 groups of order ≤ 100. For details, refer to the text preceding this table.
GAP ID Group 3 U3 U2 U2×1 A4
[1, 1] 1 7 r r r 7
[2, 1] C2 7 r r r 7
[3, 1] C3 7 r r r 7
[4, 1] C4 7 r r r 7
[4, 2] C2 × C2 7 r r r 7
[5, 1] C5 7 r r r 7
[6, 1] S3 7 4 4 4 7
[6, 2] C6 7 r r r 7
[7, 1] C7 7 r r r 7
[8, 1] C8 7 r r r 7
[8, 2] C4 × C2 7 r r r 7
GAP ID Group 3 U3 U2 U2×1 A4
[8, 4] Q8 7 4 4 4 7
[8, 5] C2 × C2× C2 7 r r r 7
[9, 1] C9 7 r r r 7
[9, 2] C3 × C3 7 r r r 7
[10, 1] D5 7 4 4 4 7
[10, 2] C10 7 r r r 7
[11, 1] C11 7 r r r 7
[12, 1] C3 oϕ C4 7 4 4 4 7
[12, 2] C12 7 r r r 7
[12, 3] A4 4 4 7 7 4
[12, 4] D6 7 4 4 4 7
[12, 5] C6 × C2 7 r r r 7
[13, 1] C13 7 r r r 7
[14, 1] D7 7 4 4 4 7
[14, 2] C14 7 r r r 7
[15, 1] C15 7 r r r 7
[16, 1] C16 7 r r r 7
[16, 2] C4 × C4 7 r r r 7
[16, 3] (C4× C2) oϕC2 7 4 7 4 7
[16, 4] C4 oϕ C4 7 4 7 4 7
[16, 5] C8 × C2 7 r r r 7
[16, 6] C8 oϕ C2 7 4 4 4 7
[16, 7] D8 7 4 4 4 7
[16, 8] QD8 7 4 4 4 7
[16, 9] Q16 7 4 4 4 7
[16, 10] C4 × C2× C2 7 r r r 7
[16, 11] C2 × D4 7 4 7 4 7
[16, 12] C2 × Q8 7 4 7 4 7
[16, 13] (C4× C2) oϕC2 7 4 4 4 7
[16, 14] C2 × C2× C2× C2 7 r r r 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[17, 1] C17 7 r r r 7
[18, 1] D9 7 4 4 4 7
[18, 2] C18 7 r r r 7
[18, 3] C3 × S3 7 4 4 4 7
[18, 4] (C3× C3) oϕC2 7 7 7 7 7
[18, 5] C6 × C3 7 r r r 7
[19, 1] C19 7 r r r 7
[20, 1] C5 oϕ C4 7 4 4 4 7
[20, 2] C20 7 r r r 7
[20, 3] C5 oϕ C4 7 7 7 7 7
[20, 4] D10 7 4 4 4 7
[20, 5] C10× C2 7 r r r 7
[21, 1] C7 oϕ C3 4 4 7 7 7
[21, 2] C21 7 r r r 7
[22, 1] D11 7 4 4 4 7
[22, 2] C22 7 r r r 7
[23, 1] C23 7 r r r 7
[24, 1] C3 oϕ C8 7 4 4 4 7
[24, 2] C24 7 r r r 7
[24, 3] SL(2, 3) 4 4 4 4 7
[24, 4] C3 oϕ Q8 7 4 4 4 7
[24, 5] C4 × S3 7 4 4 4 7
[24, 6] D12 7 4 4 4 7
[24, 7] C2 × (C3oϕC4) 7 4 7 4 7
[24, 8] (C6× C2) oϕC2 7 4 4 4 7
[24, 9] C12× C2 7 r r r 7
[24, 10] C3 × D4 7 4 4 4 7
[24, 11] C3 × Q8 7 4 4 4 7
[24, 12] S4 4 4 7 7 4
[24, 13] C2 × A4 4 4 7 7 4
GAP ID Group 3 U3 U2 U2×1 A4
[24, 14] C2 × C2× S3 7 4 7 4 7
[24, 15] C6 × C2× C2 7 r r r 7
[25, 1] C25 7 r r r 7
[25, 2] C5 × C5 7 r r r 7
[26, 1] D13 7 4 4 4 7
[26, 2] C26 7 r r r 7
[27, 1] C27 7 r r r 7
[27, 2] C9 × C3 7 r r r 7
[27, 3] (C3× C3) oϕC3 4 4 7 7 7
[27, 4] C9 oϕ C3 4 4 7 7 7
[27, 5] C3 × C3× C3 7 r r r 7
[28, 1] C7 oϕ C4 7 4 4 4 7
[28, 2] C28 7 r r r 7
[28, 3] D14 7 4 4 4 7
[28, 4] C14× C2 7 r r r 7
[29, 1] C29 7 r r r 7
[30, 1] C5 × S3 7 4 4 4 7
[30, 2] C3 × D5 7 4 4 4 7
[30, 3] D15 7 4 4 4 7
[30, 4] C30 7 r r r 7
[31, 1] C31 7 r r r 7
[32, 1] C32 7 r r r 7
[32, 2] (C4× C2) oϕC4 7 7 7 7 7
[32, 3] C8 × C4 7 r r r 7
[32, 4] C8 oϕ C4 7 4 7 4 7
[32, 5] (C8× C2) oϕC2 7 4 7 4 7
[32, 6] ((C4× C2) oϕC2) oϕC2 7 7 7 7 7
[32, 7] (C8oϕC2) oϕC2 7 7 7 7 7
[32, 8] (C2× C2) → G → (C4× C2) 7 7 7 7 7
[32, 9] (C8× C2) oϕC2 7 4 7 4 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[32, 10] Q8oϕC4 7 4 7 4 7
[32, 11] (C4× C4) oϕC2 7 4 4 4 7
[32, 12] C4 oϕ C8 7 4 7 4 7
[32, 13] C8 oϕ C4 7 4 7 4 7
[32, 14] C8 oϕ C4 7 4 7 4 7
[32, 15] C4 → G → (C4× C2) 7 4 4 4 7
[32, 16] C16× C2 7 r r r 7
[32, 17] C16oϕC2 7 4 4 4 7
[32, 18] D16 7 4 4 4 7
[32, 19] QD16 7 4 4 4 7
[32, 20] Q32 7 4 4 4 7
[32, 21] C4 × C4× C2 7 r r r 7
[32, 22] C2 × ((C4× C2) oϕC2) 7 7 7 7 7
[32, 23] C2 × (C4oϕC4) 7 7 7 7 7
[32, 24] (C4× C4) oϕC2 7 4 7 4 7
[32, 25] C4 × D4 7 4 7 4 7
[32, 26] C4 × Q8 7 4 7 4 7
[32, 27] (C2× C2× C2 × C2) oϕC2 7 7 7 7 7
[32, 28] (C4× C2× C2) oϕC2 7 7 7 7 7
[32, 29] (C2× Q8) oϕC2 7 7 7 7 7
[32, 30] (C4× C2× C2) oϕC2 7 7 7 7 7
[32, 31] (C4× C4) oϕC2 7 7 7 7 7
[32, 32] (C2× C2) → G → (C2× C2× C2) 7 7 7 7 7
[32, 33] (C4× C4) oϕC2 7 7 7 7 7
[32, 34] (C4× C4) oϕC2 7 7 7 7 7
[32, 35] C4 oϕ Q8 7 7 7 7 7
[32, 36] C8 × C2× C2 7 r r r 7
[32, 37] C2 × (C8oϕC2) 7 4 7 4 7
[32, 38] (C8× C2) oϕC2 7 4 4 4 7
[32, 39] C2 × D8 7 4 7 4 7
GAP ID Group 3 U3 U2 U2×1 A4
[32, 40] C2 × QD8 7 4 7 4 7
[32, 41] C2 × Q16 7 4 7 4 7
[32, 42] (C8× C2) oϕC2 7 4 4 4 7
[32, 43] (C2× D4) oϕC2 7 7 7 7 7
[32, 44] (C2× Q8) oϕC2 7 7 7 7 7
[32, 45] C4 × C2× C2× C2 7 r r r 7
[32, 46] C2 × C2× D4 7 7 7 7 7
[32, 47] C2 × C2× Q8 7 7 7 7 7
[32, 48] C2 × ((C4× C2) oϕC2) 7 4 7 4 7
[32, 49] (C2× D4) oϕC2 7 7 7 7 7
[32, 50] (C2× Q8) oϕC2 7 7 7 7 7
[32, 51] C2 × C2× C2× C2× C2 7 r r r 7
[33, 1] C33 7 r r r 7
[34, 1] D17 7 4 4 4 7
[34, 2] C34 7 r r r 7
[35, 1] C35 7 r r r 7
[36, 1] C9 oϕ C4 7 4 4 4 7
[36, 2] C36 7 r r r 7
[36, 3] (C2× C2) oϕC9 4 4 7 7 7
[36, 4] D18 7 4 4 4 7
[36, 5] C18× C2 7 r r r 7
[36, 6] C3 × (C3oϕC4) 7 4 4 4 7
[36, 7] (C3× C3) oϕC4 7 7 7 7 7
[36, 8] C12× C3 7 r r r 7
[36, 9] (C3× C3) oϕC4 7 7 7 7 7
[36, 10] S3× S3 7 7 7 7 7
[36, 11] C3 × A4 4 4 7 7 4
[36, 12] C6 × S3 7 4 4 4 7
[36, 13] C2 × ((C3× C3) oϕC2) 7 7 7 7 7
[36, 14] C6 × C6 7 r r r 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[37, 1] C37 7 r r r 7
[38, 1] D19 7 4 4 4 7
[38, 2] C38 7 r r r 7
[39, 1] C13oϕC3 4 4 7 7 7
[39, 2] C39 7 r r r 7
[40, 1] C5 oϕ C8 7 4 4 4 7
[40, 2] C40 7 r r r 7
[40, 3] C5 oϕ C8 7 7 7 7 7
[40, 4] C5 oϕ Q8 7 4 4 4 7
[40, 5] C4 × D5 7 4 4 4 7
[40, 6] D20 7 4 4 4 7
[40, 7] C2 × (C5oϕC4) 7 4 7 4 7
[40, 8] (C10× C2) oϕC2 7 4 4 4 7
[40, 9] C20× C2 7 r r r 7
[40, 10] C5 × D4 7 4 4 4 7
[40, 11] C5 × Q8 7 4 4 4 7
[40, 12] C2 × (C5oϕC4) 7 7 7 7 7
[40, 13] C2 × C2× D5 7 4 7 4 7
[40, 14] C10× C2× C2 7 r r r 7
[41, 1] C41 7 r r r 7
[42, 1] (C7oϕC3) oϕC2 7 7 7 7 7
[42, 2] C2 × (C7oϕC3) 4 4 7 7 7
[42, 3] C7 × S3 7 4 4 4 7
[42, 4] C3 × D7 7 4 4 4 7
[42, 5] D21 7 4 4 4 7
[42, 6] C42 7 r r r 7
[43, 1] C43 7 r r r 7
[44, 1] C11oϕC4 7 4 4 4 7
[44, 2] C44 7 r r r 7
[44, 3] D22 7 4 4 4 7
GAP ID Group 3 U3 U2 U2×1 A4
[44, 4] C22× C2 7 r r r 7
[45, 1] C45 7 r r r 7
[45, 2] C15× C3 7 r r r 7
[46, 1] D23 7 4 4 4 7
[46, 2] C46 7 r r r 7
[47, 1] C47 7 r r r 7
[48, 1] C3 oϕ C16 7 4 4 4 7
[48, 2] C48 7 r r r 7
[48, 3] (C4× C4) oϕC3 4 4 7 7 4
[48, 4] C8 × S3 7 4 4 4 7
[48, 5] C24oϕC2 7 4 4 4 7
[48, 6] C24oϕC2 7 4 4 4 7
[48, 7] D24 7 4 4 4 7
[48, 8] C3 oϕ Q16 7 4 4 4 7
[48, 9] C2 × (C3oϕC8) 7 4 7 4 7
[48, 10] (C3oϕC8) oϕC2 7 4 4 4 7
[48, 11] C4 × (C3oϕC4) 7 4 7 4 7
[48, 12] (C3oϕC4) oϕC4 7 4 7 4 7
[48, 13] C12oϕC4 7 4 7 4 7
[48, 14] (C12× C2) oϕC2 7 4 7 4 7
[48, 15] (C3× D4) oϕC2 7 7 7 7 7
[48, 16] (C3oϕC8) oϕC2 7 7 7 7 7
[48, 17] (C3× Q8) oϕC2 7 7 7 7 7
[48, 18] C3 oϕ Q16 7 7 7 7 7
[48, 19] (C2× (C3oϕC4)) oϕC2 7 4 7 4 7
[48, 20] C12× C4 7 r r r 7
[48, 21] C3 × ((C4× C2) oϕC2) 7 4 7 4 7
[48, 22] C3 × (C4oϕC4) 7 4 7 4 7
[48, 23] C24× C2 7 r r r 7
[48, 24] C3 × (C8oϕC2) 7 4 4 4 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[48, 25] C3 × D8 7 4 4 4 7
[48, 26] C3 × QD8 7 4 4 4 7
[48, 27] C3 × Q16 7 4 4 4 7
[48, 28] SL(2, 3) → G → C2 4 4 4 4 7
[48, 29] GL(2, 3) 4 4 4 4 7
[48, 30] A4oϕC4 4 4 7 7 4
[48, 31] C4 × A4 4 4 7 7 4
[48, 32] C2 × SL(2, 3) 4 4 7 4 7
[48, 33] SL(2, 3) oϕC2 4 4 4 4 7
[48, 34] C2 × (C3oϕQ8) 7 4 7 4 7
[48, 35] C2 × C4× S3 7 4 7 4 7
[48, 36] C2 × D12 7 4 7 4 7
[48, 37] (C12× C2) oϕC2 7 4 4 4 7
[48, 38] D4 × S3 7 7 7 7 7
[48, 39] (C2× (C3oϕC4)) oϕC2 7 7 7 7 7
[48, 40] Q8× S3 7 7 7 7 7
[48, 41] (C4× S3) oϕC2 7 7 7 7 7
[48, 42] C2 × C2× (C3 oϕC4) 7 7 7 7 7
[48, 43] C2 × ((C6× C2) oϕC2) 7 4 7 4 7
[48, 44] C12× C2× C2 7 r r r 7
[48, 45] C6 × D4 7 4 7 4 7
[48, 46] C6 × Q8 7 4 7 4 7
[48, 47] C3 × ((C4× C2) oϕC2) 7 4 4 4 7
[48, 48] C2 × S4 4 4 7 7 4
[48, 49] C2 × C2× A4 4 7 7 7 4
[48, 50] (C2× C2× C2 × C2) oϕC3 4 7 7 7 4
[48, 51] C2 × C2× C2× S3 7 7 7 7 7
[48, 52] C6 × C2× C2× C2 7 r r r 7
[49, 1] C49 7 r r r 7
[49, 2] C7 × C7 7 r r r 7
GAP ID Group 3 U3 U2 U2×1 A4
[50, 1] D25 7 4 4 4 7
[50, 2] C50 7 r r r 7
[50, 3] C5 × D5 7 4 4 4 7
[50, 4] (C5× C5) oϕC2 7 7 7 7 7
[50, 5] C10× C5 7 r r r 7
[51, 1] C51 7 r r r 7
[52, 1] C13oϕC4 7 4 4 4 7
[52, 2] C52 7 r r r 7
[52, 3] C13oϕC4 7 7 7 7 7
[52, 4] D26 7 4 4 4 7
[52, 5] C26× C2 7 r r r 7
[53, 1] C53 7 r r r 7
[54, 1] D27 7 4 4 4 7
[54, 2] C54 7 r r r 7
[54, 3] C3 × D9 7 4 4 4 7
[54, 4] C9 × S3 7 4 4 4 7
[54, 5] ((C3× C3) oϕC3) oϕC2 7 7 7 7 7
[54, 6] (C9oϕC3) oϕC2 7 7 7 7 7
[54, 7] (C9× C3) oϕC2 7 7 7 7 7
[54, 8] ((C3× C3) oϕC3) oϕC2 4 4 7 7 7
[54, 9] C18× C3 7 r r r 7
[54, 10] C2 × ((C3× C3) oϕC3) 4 4 7 7 7
[54, 11] C2 × (C9oϕC3) 4 4 7 7 7
[54, 12] C3 × C3× S3 7 4 7 4 7
[54, 13] C3 × ((C3× C3) oϕC2) 7 7 7 7 7
[54, 14] (C3× C3× C3) oϕC2 7 7 7 7 7
[54, 15] C6 × C3× C3 7 r r r 7
[55, 1] C11oϕC5 7 7 7 7 7
[55, 2] C55 7 r r r 7
[56, 1] C7 oϕ C8 7 4 4 4 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[56, 2] C56 7 r r r 7
[56, 3] C7 oϕ Q8 7 4 4 4 7
[56, 4] C4 × D7 7 4 4 4 7
[56, 5] D28 7 4 4 4 7
[56, 6] C2 × (C7oϕC4) 7 4 7 4 7
[56, 7] (C14× C2) oϕC2 7 4 4 4 7
[56, 8] C28× C2 7 r r r 7
[56, 9] C7 × D4 7 4 4 4 7
[56, 10] C7 × Q8 7 4 4 4 7
[56, 11] (C2× C2× C2) oϕC7 7 7 7 7 7
[56, 12] C2 × C2× D7 7 4 7 4 7
[56, 13] C14× C2× C2 7 r r r 7
[57, 1] C19oϕC3 4 4 7 7 7
[57, 2] C57 7 r r r 7
[58, 1] D29 7 4 4 4 7
[58, 2] C58 7 r r r 7
[59, 1] C59 7 r r r 7
[60, 1] C5 × (C3oϕC4) 7 4 4 4 7
[60, 2] C3 × (C5oϕC4) 7 4 4 4 7
[60, 3] C15oϕC4 7 4 4 4 7
[60, 4] C60 7 r r r 7
[60, 5] A5 4 4 7 7 4
[60, 6] C3 × (C5oϕC4) 7 7 7 7 7
[60, 7] C15oϕC4 7 7 7 7 7
[60, 8] S3× D5 7 7 7 7 7
[60, 9] C5 × A4 4 4 7 7 4
[60, 10] C6 × D5 7 4 4 4 7
[60, 11] C10× S3 7 4 4 4 7
[60, 12] D30 7 4 4 4 7
[60, 13] C30× C2 7 r r r 7
GAP ID Group 3 U3 U2 U2×1 A4
[61, 1] C61 7 r r r 7
[62, 1] D31 7 4 4 4 7
[62, 2] C62 7 r r r 7
[63, 1] C7 oϕ C9 4 4 7 7 7
[63, 2] C63 7 r r r 7
[63, 3] C3 × (C7oϕC3) 4 4 7 7 7
[63, 4] C21× C3 7 r r r 7
[64, 1] C64 7 r r r 7
[64, 2] C8 × C8 7 r r r 7
[64, 3] C8 oϕ C8 7 4 7 4 7
[64, 4] ((C8× C2) oϕC2) oϕC2 7 7 7 7 7
[64, 5] (C4× C2) oϕC8 7 7 7 7 7
[64, 6] (C8× C4) oϕC2 7 4 7 4 7
[64, 7] Q8oϕC8 7 4 7 4 7
[64, 8] ((C8× C2) oϕC2) oϕC2 7 7 7 7 7
[64, 9] (C2× Q8) oϕC4 7 7 7 7 7
[64, 10] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 11] (C4× C2) → G → (C4× C2) 7 7 7 7 7
[64, 12] (C4oϕC8) oϕC2 7 7 7 7 7
[64, 13] (C4× C2) → G → (C4× C2) 7 7 7 7 7
[64, 14] (C4× C2) → G → (C4× C2) 7 7 7 7 7
[64, 15] C8 oϕ C8 7 4 7 4 7
[64, 16] C8 oϕ C8 7 4 7 4 7
[64, 17] (C8× C2) oϕC4 7 7 7 7 7
[64, 18] (C8× C2) oϕC4 7 7 7 7 7
[64, 19] C4 → G → (C4× C4) 7 7 7 7 7
[64, 20] (C4× C4) oϕC4 7 4 7 4 7
[64, 21] (C8× C2) oϕC4 7 7 7 7 7
[64, 22] C4 → G → (C4× C4) 7 4 7 4 7
[64, 23] (C4× C2× C2) oϕC4 7 7 7 7 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[64, 24] (C8oϕC2) oϕC4 7 7 7 7 7
[64, 25] (C8× C2) oϕC4 7 7 7 7 7
[64, 26] C16× C4 7 r r r 7
[64, 27] C16oϕC4 7 4 7 4 7
[64, 28] C16oϕC4 7 7 7 7 7
[64, 29] (C16× C2) oϕC2 7 4 7 4 7
[64, 30] (C16oϕC2) oϕC2 7 7 7 7 7
[64, 31] (C16× C2) oϕC2 7 4 4 4 7
[64, 32] ((C8oϕC2) oϕC2) oϕC2 7 7 7 7 7
[64, 33] (C4× C2× C2) oϕC4 7 7 7 7 7
[64, 34] (((C4× C2) oϕC2) oϕC2) oϕC2 7 7 7 7 7
[64, 35] (C4× C4) oϕC4 7 7 7 7 7
[64, 36] (C2× C2) → G → (C4× C2)) oϕC2 7 7 7 7 7
[64, 37] (C4× C2) → G → (C4× C2) 7 7 7 7 7
[64, 38] (C16× C2) oϕC2 7 4 7 4 7
[64, 39] Q16oϕC4 7 4 7 4 7
[64, 40] (C16× C2) oϕC2 7 4 4 4 7
[64, 41] (C16oϕC2) oϕC2 7 7 7 7 7
[64, 42] (C16oϕC2) oϕC2 7 7 7 7 7
[64, 43] C8 → G → (C4× C2) 7 7 7 7 7
[64, 44] C4 oϕ C16 7 4 7 4 7
[64, 45] C4 → G → (C8× C2) 7 4 4 4 7
[64, 46] C16oϕC4 7 7 7 7 7
[64, 47] C16oϕC4 7 4 7 4 7
[64, 48] C16oϕC4 7 4 7 4 7
[64, 49] C8 → G → (C4× C2) 7 4 4 4 7
[64, 50] C32× C2 7 r r r 7
[64, 51] C32oϕC2 7 4 4 4 7
[64, 52] D32 7 4 4 4 7
[64, 53] QD32 7 4 4 4 7
GAP ID Group 3 U3 U2 U2×1 A4
[64, 54] Q64 7 4 4 4 7
[64, 55] C4 × C4× C4 7 r r r 7
[64, 56] C2 × ((C4× C2) oϕC4) 7 7 7 7 7
[64, 57] (C4× C4) oϕC4 7 7 7 7 7
[64, 58] C4 × ((C4× C2) oϕC2) 7 7 7 7 7
[64, 59] C4 × (C4oϕC4) 7 7 7 7 7
[64, 60] (C2× ((C4× C2) oϕC2)) oϕC2 7 7 7 7 7
[64, 61] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 62] ((C4× C2) oϕC4) oϕC2 7 7 7 7 7
[64, 63] (C4× C4) oϕC4 7 7 7 7 7
[64, 64] (C4× C4) oϕC4 7 7 7 7 7
[64, 65] (C4× C4) oϕC4 7 7 7 7 7
[64, 66] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 67] (C4× C2× C2 × C2) oϕC2 7 7 7 7 7
[64, 68] (C4oϕC4) oϕC4 7 7 7 7 7
[64, 69] (C4× C4× C2) oϕC2 7 7 7 7 7
[64, 70] (C4oϕC4) oϕC4 7 7 7 7 7
[64, 71] (C4× C4× C2) oϕC2 7 7 7 7 7
[64, 72] (C2× Q8) oϕC4 7 7 7 7 7
[64, 73] (C2× C2× D4) oϕC2 7 7 7 7 7
[64, 74] (C2× C2× Q8) oϕC2 7 7 7 7 7
[64, 75] (C2× ((C4× C2) oϕC2)) oϕC2 7 7 7 7 7
[64, 76] (C4× C2) oϕQ8 7 7 7 7 7
[64, 77] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 78] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 79] (C2× C2× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 80] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 81] (C2× C2× C2) → G → (C2× C2× C2) 7 7 7 7 7 [64, 82] (C2× C2× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 83] C8 × C4× C2 7 r r r 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[64, 84] C2 × (C8oϕC4) 7 7 7 7 7
[64, 85] C4 × (C8oϕC2) 7 4 7 4 7
[64, 86] (C8× C4) oϕC2 7 4 7 4 7
[64, 87] C2 × ((C8× C2) oϕC2) 7 7 7 7 7
[64, 88] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 89] (C8× C2× C2) oϕC2 7 7 7 7 7
[64, 90] C2 × (((C4× C2) oϕC2) oϕC2) 7 7 7 7 7 [64, 91] (((C4× C2) oϕC2) oϕC2) oϕC2 7 7 7 7 7
[64, 92] C2 × ((C8oϕC2) oϕC2) 7 7 7 7 7
[64, 93] (C2× C2) → G → (C4× C2)) 7 7 7 7 7
[64, 94] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 95] C2 × ((C8× C2) oϕC2) 7 7 7 7 7
[64, 96] C2 × (Q8oϕC4) 7 7 7 7 7
[64, 97] (C8× C2× C2) oϕC2 7 4 7 4 7
[64, 98] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 99] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 100] (Q8oϕC4) oϕC2 7 7 7 7 7
[64, 101] C2 × ((C4× C4) oϕC2) 7 4 7 4 7
[64, 102] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 103] C2 × (C4oϕC8) 7 7 7 7 7
[64, 104] (C4oϕC8) oϕC2 7 7 7 7 7
[64, 105] (C4oϕC8) oϕC2 7 7 7 7 7
[64, 106] C2 × (C8oϕC4) 7 7 7 7 7
[64, 107] C2 × (C8oϕC4) 7 7 7 7 7
[64, 108] (C8oϕC4) oϕC2 7 4 7 4 7
[64, 109] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 110] C4 → G → (C4× C2)) 7 4 7 4 7
[64, 111] C4 → G → (C4× C2)) oϕC2 7 7 7 7 7
[64, 112] (C8× C4) oϕC2 7 4 7 4 7
[64, 113] (C4oϕC8) oϕC2 7 7 7 7 7
GAP ID Group 3 U3 U2 U2×1 A4
[64, 114] (C8× C4) oϕC2 7 7 7 7 7
[64, 115] C8 × D4 7 4 7 4 7
[64, 116] (C8× C2× C2) oϕC2 7 7 7 7 7
[64, 117] (C8× C4) oϕC2 7 7 7 7 7
[64, 118] C4 × D8 7 4 7 4 7
[64, 119] C4 × QD8 7 4 7 4 7
[64, 120] C4 × Q16 7 4 7 4 7
[64, 121] (C4× Q8) oϕC2 7 7 7 7 7
[64, 122] Q16oϕC4 7 7 7 7 7
[64, 123] (C4× D4) oϕC2 7 7 7 7 7
[64, 124] (C8× C4) oϕC2 7 4 4 4 7
[64, 125] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 126] C8 × Q8 7 4 7 4 7
[64, 127] C8 oϕ Q8 7 7 7 7 7
[64, 128] (C2× C2× D4) oϕC2 7 7 7 7 7
[64, 129] (C2× C2× Q8) oϕC2 7 7 7 7 7
[64, 130] (C2× D8) oϕC2 7 7 7 7 7
[64, 131] (C2× QD8) oϕC2 7 7 7 7 7
[64, 132] (C2× Q16) oϕC2 7 7 7 7 7
[64, 133] (C2× Q16) oϕC2 7 7 7 7 7
[64, 134] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 135] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 136] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 137] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 138] (((C4× C2) oϕC2) oϕC2) oϕC2 7 7 7 7 7 [64, 139] (((C4× C2) oϕC2) oϕC2) oϕC2 7 7 7 7 7
[64, 140] (C4× D4) oϕC2 7 7 7 7 7
[64, 141] (C2× QD8) oϕC2 7 7 7 7 7
[64, 142] (Q8oϕC4) oϕC2 7 7 7 7 7
[64, 143] C4 oϕ Q16 7 7 7 7 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[64, 144] (C4× D4) oϕC2 7 7 7 7 7
[64, 145] (C2× Q16) oϕC2 7 7 7 7 7
[64, 146] (C8× C2× C2) oϕC2 7 7 7 7 7
[64, 147] (C8× C2× C2) oϕC2 7 7 7 7 7
[64, 148] (C2× Q16) oϕC2 7 7 7 7 7
[64, 149] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 150] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 151] (C2× Q16) oϕC2 7 7 7 7 7
[64, 152] (C2× QD8) oϕC2 7 7 7 7 7
[64, 153] (C2× D8) oϕC2 7 7 7 7 7
[64, 154] (C2× Q16) oϕC2 7 7 7 7 7
[64, 155] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 156] Q8oϕQ8 7 7 7 7 7
[64, 157] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 158] Q8oϕQ8 7 7 7 7 7
[64, 159] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 160] (C4× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 161] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 162] (C2× (C4oϕC4)) oϕC2 7 7 7 7 7
[64, 163] ((C8× C2) oϕC2) oϕC2 7 7 7 7 7
[64, 164] (Q8oϕC4) oϕC2 7 7 7 7 7
[64, 165] (Q8oϕC4) oϕC2 7 7 7 7 7
[64, 166] (C8oϕC4) oϕC2 7 7 7 7 7
[64, 167] (C8× C4) oϕC2 7 7 7 7 7
[64, 168] (C4× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 169] (C8× C4) oϕC2 7 7 7 7 7
[64, 170] (Q8oϕC4) oϕC2 7 7 7 7 7
[64, 171] ((C8× C2) oϕC2) oϕC2 7 7 7 7 7
[64, 172] (C4× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 173] (C8× C4) oϕC2 7 7 7 7 7
GAP ID Group 3 U3 U2 U2×1 A4
[64, 174] (C8× C4) oϕC2 7 7 7 7 7
[64, 175] C4 oϕ Q16 7 7 7 7 7
[64, 176] (C8× C4) oϕC2 7 7 7 7 7
[64, 177] (C2× D8) oϕC2 7 7 7 7 7
[64, 178] (C2× Q16) oϕC2 7 7 7 7 7
[64, 179] C8 oϕ Q8 7 7 7 7 7
[64, 180] (C4× C2) → G → (C2× C2× C2) 7 7 7 7 7
[64, 181] C8 oϕ Q8 7 7 7 7 7
[64, 182] C8 oϕ Q8 7 7 7 7 7
[64, 183] C16× C2× C2 7 r r r 7
[64, 184] C2 × (C16oϕC2) 7 4 7 4 7
[64, 185] (C16× C2) oϕC2 7 4 4 4 7
[64, 186] C2 × D16 7 4 7 4 7
[64, 187] C2 × QD16 7 4 7 4 7
[64, 188] C2 × Q32 7 4 7 4 7
[64, 189] (C16× C2) oϕC2 7 4 4 4 7
[64, 190] (C2× D8) oϕC2 7 7 7 7 7
[64, 191] (C2× Q16) oϕC2 7 7 7 7 7
[64, 192] C4 × C4× C2× C2 7 r r r 7
[64, 193] C2 × C2× ((C4× C2) oϕC2) 7 7 7 7 7
[64, 194] C2 × C2× (C4 oϕC4) 7 7 7 7 7
[64, 195] C2 × ((C4× C4) oϕC2) 7 7 7 7 7
[64, 196] C2 × C4× D4 7 7 7 7 7
[64, 197] C2 × C4× Q8 7 7 7 7 7
[64, 198] C4 × ((C4× C2) oϕC2) 7 4 7 4 7
[64, 199] (C4× D4) oϕC2 7 7 7 7 7
[64, 200] (C4× Q8) oϕC2 7 7 7 7 7
[64, 201] (C4× Q8) oϕC2 7 7 7 7 7
[64, 202] C2 × ((C2× C2× C2× C2) oϕC2) 7 7 7 7 7 [64, 203] C2 × ((C4× C2× C2) oϕC2) 7 7 7 7 7
Continued on next page
GAP ID Group 3 U3 U2 U2×1 A4
[64, 204] C2 × ((C2× Q8) oϕC2) 7 7 7 7 7
[64, 205] C2 × ((C4× C2× C2) oϕC2) 7 7 7 7 7
[64, 206] (C4× C2× C2 × C2) oϕC2 7 7 7 7 7
[64, 207] C2 × ((C4× C4) oϕC2) 7 7 7 7 7
[64, 208] C2 × ((C2× C2) → G → (C2× C2× C2)) 7 7 7 7 7
[64, 209] C2 × ((C4× C4) oϕC2) 7 7 7 7 7
[64, 210] (C4× C4× C2) oϕC2 7 7 7 7 7
[64, 211] C2 × ((C4× C4) oϕC2) 7 7 7 7 7
[64, 212] C2 × (C4oϕQ8) 7 7 7 7 7
[64, 213] (C4× C4× C2) oϕC2 7 7 7 7 7
[64, 214] (C4× Q8) oϕC2 7 7 7 7 7
[64, 215] (C2× C2× D4) oϕC2 7 7 7 7 7
[64, 216] (C2× C2× D4) oϕC2 7 7 7 7 7
[64, 217] (C2× C2× Q8) oϕC2 7 7 7 7 7
[64, 218] (C2× ((C4× C2) oϕC2)) oϕC2 7 7 7 7 7
[64, 219] (C4× D4) oϕC2 7 7 7 7 7
[64, 220] (C4× D4) oϕC2 7 7 7 7 7
[64, 221] (C4× D4) oϕC2 7 7 7 7 7
[64, 222] (C4× Q8) oϕC2 7 7 7 7 7
[64, 223] (C4× Q8) oϕC2 7 7 7 7 7
[64, 224] ((C2× Q8) oϕC2) oϕC2 7 7 7 7 7
[64, 225] (C4oϕQ8) oϕC2 7 7 7 7 7
[64, 226] D4 × D4 7 7 7 7 7
[64, 227] (C2× C2× D4) oϕC2 7 7 7 7 7
[64, 228] (C4× D4) oϕC2 7 7 7 7 7
[64, 229] (C2× C2× Q8) oϕC2 7 7 7 7 7
[64, 230] Q8× D4 7 7 7 7 7
[64, 231] (C4× D4) oϕC2 7 7 7 7 7
[64, 232] (C4× D4) oϕC2 7 7 7 7 7
[64, 233] (C4× Q8) oϕC2 7 7 7 7 7
GAP ID Group 3 U3 U2 U2×1 A4
[64, 234] (C4× D4) oϕC2 7 7 7 7 7
[64, 235] (C4× Q8) oϕC2 7 7 7 7 7
[64, 236] (C4× D4) oϕC2 7 7 7 7 7
[64, 237] (C4× Q8) oϕC2 7 7 7 7 7
[64, 238] Q8oϕQ8 7 7 7 7 7
[64, 239] Q8× Q8 7 7 7 7 7
[64, 240] (C4× D4) oϕC2 7 7 7 7 7
[64, 241] ((C4× C2× C2) oϕC2) oϕC2 7 7 7 7 7
[64, 242] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 243] ((C2× C2) → G → (C2× C2× C2)) oϕC2 7 7 7 7 7
[64, 244] ((C4× C4) oϕC2) oϕC2 7 7 7 7 7
[64, 245] (C2× C2) → G → (C2× C2× C2× C2) 7 7 7 7 7
[64, 246] C8 × C2× C2× C2 7 r r r 7
[64, 247] C2 × C2× (C8 oϕC2) 7 7 7 7 7
[64, 248] C2 × ((C8× C2) oϕC2) 7 4 7 4 7
[64, 249] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 250] C2 × C2× D8 7 7 7 7 7
[64, 251] C2 × C2× QD8 7 7 7 7 7
[64, 252] C2 × C2× Q16 7 7 7 7 7
[64, 253] C2 × ((C8× C2) oϕC2) 7 4 7 4 7
[64, 254] C2 × ((C2× D4) oϕC2) 7 7 7 7 7
[64, 255] C2 × ((C2× Q8) oϕC2) 7 7 7 7 7
[64, 256] (C2× (C8oϕC2)) oϕC2 7 7 7 7 7
[64, 257] (C2× D8) oϕC2 7 7 7 7 7
[64, 258] (C2× QD8) oϕC2 7 7 7 7 7
[64, 259] (C2× Q16) oϕC2 7 7 7 7 7
[64, 260] C4 × C2× C2× C2× C2 7 r r r 7
[64, 261] C2 × C2× C2× D4 7 7 7 7 7
[64, 262] C2 × C2× C2× Q8 7 7 7 7 7
[64, 263] C2 × C2× ((C4× C2) oϕC2) 7 7 7 7 7 Continued on next page
