p-group, metacyclic, nilpotent (class 5), monomial
Aliases: C32.1C4, C16.4Q8, C4.19D16, C8.10Q16, C22.1Q32, (C2×C32).5C2, (C2×C4).70D8, C8.17(C4⋊C4), C16.18(C2×C4), (C2×C8).266D4, C2.5(C16⋊3C4), C4.12(C2.D8), C8.4Q8.1C2, (C2×C16).93C22, SmallGroup(128,157)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32.C4
G = < a,b | a32=1, b4=a16, bab-1=a15 >
Smallest permutation representation of C32.C4
►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 52 9 44 17 36 25 60)(2 35 10 59 18 51 26 43)(3 50 11 42 19 34 27 58)(4 33 12 57 20 49 28 41)(5 48 13 40 21 64 29 56)(6 63 14 55 22 47 30 39)(7 46 15 38 23 62 31 54)(8 61 16 53 24 45 32 37)
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), (1,52,9,44,17,36,25,60)(2,35,10,59,18,51,26,43)(3,50,11,42,19,34,27,58)(4,33,12,57,20,49,28,41)(5,48,13,40,21,64,29,56)(6,63,14,55,22,47,30,39)(7,46,15,38,23,62,31,54)(8,61,16,53,24,45,32,37)>;
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), (1,52,9,44,17,36,25,60)(2,35,10,59,18,51,26,43)(3,50,11,42,19,34,27,58)(4,33,12,57,20,49,28,41)(5,48,13,40,21,64,29,56)(6,63,14,55,22,47,30,39)(7,46,15,38,23,62,31,54)(8,61,16,53,24,45,32,37) );
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)], [(1,52,9,44,17,36,25,60),(2,35,10,59,18,51,26,43),(3,50,11,42,19,34,27,58),(4,33,12,57,20,49,28,41),(5,48,13,40,21,64,29,56),(6,63,14,55,22,47,30,39),(7,46,15,38,23,62,31,54),(8,61,16,53,24,45,32,37)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | Q16 | D8 | D16 | Q32 | C32.C4 |
| kernel | C32.C4 | C8.4Q8 | C2×C32 | C32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C32.C4 ►in GL2(𝔽97) generated by
| 30 | 0 |
| 0 | 42 |
| 0 | 1 |
| 75 | 0 |
G:=sub<GL(2,GF(97))| [30,0,0,42],[0,75,1,0] >;
C32.C4 in GAP, Magma, Sage, TeX
C_{32}.C_4 % in TeX
G:=Group("C32.C4"); // GroupNames label
G:=SmallGroup(128,157);
// by ID
G=gap.SmallGroup(128,157);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,148,422,268,248,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b|a^32=1,b^4=a^16,b*a*b^-1=a^15>;
// generators/relations
Export