p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C32:2C4, C16.2C8, C42.3C8, C8.21C42, C4.6M5(2), M6(2).2C2, C22.4M5(2), (C4xC8).8C4, (C2xC8).3C8, C8.22(C2xC8), C4.20(C4xC8), (C2xC16).5C4, C16.22(C2xC4), C16:5C4.8C2, C2.4(C16:5C4), (C2xC16).48C22, (C2xC4).76(C2xC8), (C2xC8).240(C2xC4), SmallGroup(128,130)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32:C4
G = < a,b | a32=b4=1, bab-1=a9 >
Quotients: C1, C2, C4, C22, C8, C2xC4, C42, C2xC8, C4xC8, M5(2), C16:5C4, C32:C4
Smallest permutation representation of C32:C4
►On 32 points
Generators in S32
(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)
(2 26 18 10)(3 19)(4 12 20 28)(6 30 22 14)(7 23)(8 16 24 32)(11 27)(15 31)
G:=sub<Sym(32)| (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), (2,26,18,10)(3,19)(4,12,20,28)(6,30,22,14)(7,23)(8,16,24,32)(11,27)(15,31)>;
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), (2,26,18,10)(3,19)(4,12,20,28)(6,30,22,14)(7,23)(8,16,24,32)(11,27)(15,31) );
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)], [(2,26,18,10),(3,19),(4,12,20,28),(6,30,22,14),(7,23),(8,16,24,32),(11,27),(15,31)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | M5(2) | M5(2) | C32:C4 |
| kernel | C32:C4 | C16:5C4 | M6(2) | C32 | C4xC8 | C2xC16 | C16 | C42 | C2xC8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 4 |
Matrix representation of C32:C4 ►in GL4(F97) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 96 |
| 0 | 22 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 75 | 0 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(97))| [0,0,0,64,0,0,22,0,1,0,0,0,0,96,0,0],[1,0,0,0,0,96,0,0,0,0,75,0,0,0,0,22] >;
C32:C4 in GAP, Magma, Sage, TeX
C_{32}\rtimes C_4 % in TeX
G:=Group("C32:C4"); // GroupNames label
G:=SmallGroup(128,130);
// by ID
G=gap.SmallGroup(128,130);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,28,477,64,723,100,2019,102,124]);
// Polycyclic
G:=Group<a,b|a^32=b^4=1,b*a*b^-1=a^9>;
// generators/relations
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