p-group, cyclic, abelian, monomial
Aliases: C64, also denoted Z64, SmallGroup(64,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C64 |
| C1 — C64 |
| C1 — C64 |
| C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C8 — C8 — C8 — C8 — C16 — C16 — C32 — C64 |
Generators and relations for C64
G = < a | a64=1 >
Smallest permutation representation of C64
►Regular action on 64 points
Generators in S64
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
G:=sub<Sym(64)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)]])
C64 is a maximal subgroup of
C128 D64 SD128 Q128
C2p.C32: M7(2) C3⋊C64 C5⋊2C64 C5⋊C64 C7⋊C64 ...
C64 is a maximal quotient of
C128 C3⋊C64 C5⋊2C64 C5⋊C64 C7⋊C64
64 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P | 64A | ··· | 64AF |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 | 64 | ··· | 64 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | |||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 |
| kernel | C64 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 |
Matrix representation of C64 ►in GL1(𝔽193) generated by
| 154 |
G:=sub<GL(1,GF(193))| [154] >;
C64 in GAP, Magma, Sage, TeX
C_{64} % in TeX
G:=Group("C64"); // GroupNames label
G:=SmallGroup(64,1);
// by ID
G=gap.SmallGroup(64,1);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-2,12,31,50,69,88]);
// Polycyclic
G:=Group<a|a^64=1>;
// generators/relations
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