p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42:1C8, C42.1Q8, C42.19D4, (C4xC8):1C4, C4.1(C4:C8), C4.16(C4xC8), (C2xC42).4C4, (C2xC4).78C42, C4.11(C8:C4), C42:4C4.3C2, C42.288(C2xC4), (C22xC4).116D4, (C2xC4).63M4(2), C2.1(C4.9C42), C22.8(C22:C8), C42.12C4.1C2, (C2xC42).121C22, C23.133(C22:C4), C2.3(C22.7C42), C22.15(C2.C42), (C2xC4).66(C2xC8), (C2xC4).92(C4:C4), (C22xC4).460(C2xC4), (C2xC4).287(C22:C4), SmallGroup(128,6)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42:1C8
G = < a,b,c | a4=b4=c8=1, ab=ba, cac-1=ab-1, bc=cb >
Subgroups: 152 in 84 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, C23, C42, C42, C42, C2xC8, C22xC4, C22xC4, C22xC4, C2.C42, C4xC8, C22:C8, C4:C8, C2xC42, C2xC42, C42:4C4, C42.12C4, C42:1C8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, C42, C22:C4, C4:C4, C2xC8, M4(2), C2.C42, C4xC8, C8:C4, C22:C8, C4:C8, C22.7C42, C4.9C42, C42:1C8
►On 32 points
(1 5)(2 16 32 18)(3 29)(4 20 26 10)(6 12 28 22)(7 25)(8 24 30 14)(9 23)(11 15)(13 19)(17 21)(27 31)
(1 21 31 11)(2 22 32 12)(3 23 25 13)(4 24 26 14)(5 17 27 15)(6 18 28 16)(7 19 29 9)(8 20 30 10)
(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)
G:=sub<Sym(32)| (1,5)(2,16,32,18)(3,29)(4,20,26,10)(6,12,28,22)(7,25)(8,24,30,14)(9,23)(11,15)(13,19)(17,21)(27,31), (1,21,31,11)(2,22,32,12)(3,23,25,13)(4,24,26,14)(5,17,27,15)(6,18,28,16)(7,19,29,9)(8,20,30,10), (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)>;
G:=Group( (1,5)(2,16,32,18)(3,29)(4,20,26,10)(6,12,28,22)(7,25)(8,24,30,14)(9,23)(11,15)(13,19)(17,21)(27,31), (1,21,31,11)(2,22,32,12)(3,23,25,13)(4,24,26,14)(5,17,27,15)(6,18,28,16)(7,19,29,9)(8,20,30,10), (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) );
G=PermutationGroup([[(1,5),(2,16,32,18),(3,29),(4,20,26,10),(6,12,28,22),(7,25),(8,24,30,14),(9,23),(11,15),(13,19),(17,21),(27,31)], [(1,21,31,11),(2,22,32,12),(3,23,25,13),(4,24,26,14),(5,17,27,15),(6,18,28,16),(7,19,29,9),(8,20,30,10)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D4 | M4(2) | C4.9C42 |
| kernel | C42:1C8 | C42:4C4 | C42.12C4 | C4xC8 | C2xC42 | C42 | C42 | C42 | C22xC4 | C2xC4 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 | 1 | 1 | 2 | 4 | 4 |
Matrix representation of C42:1C8 ►in GL6(F17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42:1C8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_1C_8
% in TeX
G:=Group("C4^2:1C8"); // GroupNames label
G:=SmallGroup(128,6);
// by ID
G=gap.SmallGroup(128,6);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,184,570,248,1684]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
// generators/relations