p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.13C42, C23.5Q16, C8.3(C4⋊C4), (C2×C8).4Q8, (C2×C16).4C4, (C2×C8).83D4, (C2×C4).11D8, C8.C4.4C4, (C2×C4).95SD16, C4.16(C4.Q8), C8.36(C22⋊C4), (C22×C4).191D4, C4.32(D4⋊C4), (C2×M5(2)).13C2, C22.11(C2.D8), C4.6(C2.C42), (C22×C8).204C22, C22.2(Q8⋊C4), C2.15(C22.4Q16), (C2×C8).51(C2×C4), (C2×C4).113(C4⋊C4), (C2×C8.C4).4C2, (C2×C4).64(C22⋊C4), SmallGroup(128,117)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.13C42
G = < a,b,c | a8=1, b4=a4, c4=a6, bab-1=a3, ac=ca, cbc-1=a3b >
Subgroups: 104 in 62 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, M5(2), C22×C8, C2×M4(2), C2×C8.C4, C2×M5(2), C8.13C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, C8.13C42
►On 32 points
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)
(1 19 5 31 9 27 13 23)(2 26 14 30 10 18 6 22)(3 17 7 29 11 25 15 21)(4 24 16 28 12 32 8 20)
(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)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28), (1,19,5,31,9,27,13,23)(2,26,14,30,10,18,6,22)(3,17,7,29,11,25,15,21)(4,24,16,28,12,32,8,20), (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( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28), (1,19,5,31,9,27,13,23)(2,26,14,30,10,18,6,22)(3,17,7,29,11,25,15,21)(4,24,16,28,12,32,8,20), (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([[(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28)], [(1,19,5,31,9,27,13,23),(2,26,14,30,10,18,6,22),(3,17,7,29,11,25,15,21),(4,24,16,28,12,32,8,20)], [(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)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8N | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C8.13C42 |
| kernel | C8.13C42 | C2×C8.C4 | C2×M5(2) | C8.C4 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 4 |
Matrix representation of C8.13C42 ►in GL4(𝔽17) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 16 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 13 | 9 | 0 | 0 |
| 0 | 0 | 4 | 12 |
| 0 | 0 | 15 | 13 |
G:=sub<GL(4,GF(17))| [9,0,0,0,0,9,0,0,0,0,15,0,0,0,0,15],[0,0,4,0,0,0,16,13,1,0,0,0,0,1,0,0],[8,13,0,0,7,9,0,0,0,0,4,15,0,0,12,13] >;
C8.13C42 in GAP, Magma, Sage, TeX
C_8._{13}C_4^2 % in TeX
G:=Group("C8.13C4^2"); // GroupNames label
G:=SmallGroup(128,117);
// by ID
G=gap.SmallGroup(128,117);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,248,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^4,c^4=a^6,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations