p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.31D8, C8.36SD16, C4⋊C16⋊2C2, C4⋊C4.2C8, C4⋊C8.4C4, C4.41C4≀C2, (C2×D4).3C8, (C4×D4).1C4, C16⋊5C4⋊6C2, C2.5(D4⋊C8), (C2×C8).371D4, C8⋊6D4.10C2, C2.6(D4.C8), C42.42(C2×C4), (C4×C8).303C22, C2.4(C23.C8), (C2×C4).10M4(2), C4.46(D4⋊C4), C22.50(C22⋊C8), (C2×C4).12(C2×C8), (C2×C4).381(C22⋊C4), SmallGroup(128,62)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.31D8
G = < a,b,c | a8=1, b8=a4, c2=a, bab-1=a5, ac=ca, cbc-1=a5b7 >
Smallest permutation representation of C8.31D8
►On 64 points
Generators in S64
(1 53 37 26 9 61 45 18)(2 62 38 19 10 54 46 27)(3 55 39 28 11 63 47 20)(4 64 40 21 12 56 48 29)(5 57 41 30 13 49 33 22)(6 50 42 23 14 58 34 31)(7 59 43 32 15 51 35 24)(8 52 44 25 16 60 36 17)
(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 60 53 36 37 17 26 8 9 52 61 44 45 25 18 16)(2 43 62 32 38 15 19 51 10 35 54 24 46 7 27 59)(3 23 55 14 39 58 28 34 11 31 63 6 47 50 20 42)(4 5 64 57 40 41 21 30 12 13 56 49 48 33 29 22)
G:=sub<Sym(64)| (1,53,37,26,9,61,45,18)(2,62,38,19,10,54,46,27)(3,55,39,28,11,63,47,20)(4,64,40,21,12,56,48,29)(5,57,41,30,13,49,33,22)(6,50,42,23,14,58,34,31)(7,59,43,32,15,51,35,24)(8,52,44,25,16,60,36,17), (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), (1,60,53,36,37,17,26,8,9,52,61,44,45,25,18,16)(2,43,62,32,38,15,19,51,10,35,54,24,46,7,27,59)(3,23,55,14,39,58,28,34,11,31,63,6,47,50,20,42)(4,5,64,57,40,41,21,30,12,13,56,49,48,33,29,22)>;
G:=Group( (1,53,37,26,9,61,45,18)(2,62,38,19,10,54,46,27)(3,55,39,28,11,63,47,20)(4,64,40,21,12,56,48,29)(5,57,41,30,13,49,33,22)(6,50,42,23,14,58,34,31)(7,59,43,32,15,51,35,24)(8,52,44,25,16,60,36,17), (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), (1,60,53,36,37,17,26,8,9,52,61,44,45,25,18,16)(2,43,62,32,38,15,19,51,10,35,54,24,46,7,27,59)(3,23,55,14,39,58,28,34,11,31,63,6,47,50,20,42)(4,5,64,57,40,41,21,30,12,13,56,49,48,33,29,22) );
G=PermutationGroup([[(1,53,37,26,9,61,45,18),(2,62,38,19,10,54,46,27),(3,55,39,28,11,63,47,20),(4,64,40,21,12,56,48,29),(5,57,41,30,13,49,33,22),(6,50,42,23,14,58,34,31),(7,59,43,32,15,51,35,24),(8,52,44,25,16,60,36,17)], [(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)], [(1,60,53,36,37,17,26,8,9,52,61,44,45,25,18,16),(2,43,62,32,38,15,19,51,10,35,54,24,46,7,27,59),(3,23,55,14,39,58,28,34,11,31,63,6,47,50,20,42),(4,5,64,57,40,41,21,30,12,13,56,49,48,33,29,22)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | ··· | 8H | 8I | 8J | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D8 | SD16 | M4(2) | C4≀C2 | D4.C8 | C23.C8 |
| kernel | C8.31D8 | C16⋊5C4 | C4⋊C16 | C8⋊6D4 | C4⋊C8 | C4×D4 | C4⋊C4 | C2×D4 | C2×C8 | C8 | C8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 2 |
Matrix representation of C8.31D8 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
| 0 | 12 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 13 | 2 |
| 0 | 0 | 8 | 4 |
| 0 | 12 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 13 | 2 |
| 0 | 0 | 9 | 13 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,13,0,0,1,0],[0,11,0,0,12,12,0,0,0,0,13,8,0,0,2,4],[0,6,0,0,12,0,0,0,0,0,13,9,0,0,2,13] >;
C8.31D8 in GAP, Magma, Sage, TeX
C_8._{31}D_8 % in TeX
G:=Group("C8.31D8"); // GroupNames label
G:=SmallGroup(128,62);
// by ID
G=gap.SmallGroup(128,62);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,891,436,1018,136,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^8=a^4,c^2=a,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^5*b^7>;
// generators/relations
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