direct product, p-group, abelian, monomial
Aliases: C82, SmallGroup(64,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C82 |
| C1 — C82 |
| C1 — C82 |
Generators and relations for C82
G = < a,b | a8=b8=1, ab=ba >
Smallest permutation representation of C82
►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)
(1 42 34 12 57 29 55 17)(2 43 35 13 58 30 56 18)(3 44 36 14 59 31 49 19)(4 45 37 15 60 32 50 20)(5 46 38 16 61 25 51 21)(6 47 39 9 62 26 52 22)(7 48 40 10 63 27 53 23)(8 41 33 11 64 28 54 24)
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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24)>;
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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24) );
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)], [(1,42,34,12,57,29,55,17),(2,43,35,13,58,30,56,18),(3,44,36,14,59,31,49,19),(4,45,37,15,60,32,50,20),(5,46,38,16,61,25,51,21),(6,47,39,9,62,26,52,22),(7,48,40,10,63,27,53,23),(8,41,33,11,64,28,54,24)]])
C82 is a maximal subgroup of
C16⋊5C8 C8⋊C16 C8≀C2 C8⋊2C16 C8.36D8 C82⋊C2 C82⋊15C2 C82⋊2C2 C8⋊6M4(2) C8⋊6D8 C8⋊9SD16 C8⋊6Q16 C8⋊8SD16 C8⋊5D8 C8⋊5Q16 C82⋊12C2 C82⋊5C2 C8.7Q16 C82⋊3C2 C8⋊4D8 C8⋊4Q16 C82⋊C3
C82 is a maximal quotient of
C2.C82 C16⋊5C8
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C4 | C8 |
| kernel | C82 | C4×C8 | C2×C8 | C8 |
| # reps | 1 | 3 | 12 | 48 |
Matrix representation of C82 ►in GL2(𝔽17) generated by
| 8 | 0 |
| 0 | 13 |
| 13 | 0 |
| 0 | 15 |
G:=sub<GL(2,GF(17))| [8,0,0,13],[13,0,0,15] >;
C82 in GAP, Magma, Sage, TeX
C_8^2
% in TeX
G:=Group("C8^2"); // GroupNames label
G:=SmallGroup(64,2);
// by ID
G=gap.SmallGroup(64,2);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,24,55,86,117]);
// Polycyclic
G:=Group<a,b|a^8=b^8=1,a*b=b*a>;
// generators/relations
Export