p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.4C42, M5(2)⋊10C4, C23.14SD16, (C2×C8).8Q8, C8.29(C4⋊C4), C8.C4⋊4C4, C4.Q8.4C4, (C2×C4).125D8, (C2×C8).355D4, (C2×C4).10Q16, C4.4(C2.D8), (C2×C4).96SD16, C8.38(C22⋊C4), C4.8(Q8⋊C4), (C22×C4).195D4, C4.52(D4⋊C4), C22.4(C4.Q8), (C2×M5(2)).17C2, (C22×C8).208C22, C22.23(D4⋊C4), C2.19(C22.4Q16), C4.10(C2.C42), C22.10(Q8⋊C4), C23.25D4.10C2, (C2×C8).55(C2×C4), (C2×C4).31(C4⋊C4), (C2×C8.C4).7C2, (C2×C4).233(C22⋊C4), SmallGroup(128,121)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.4C42
G = < a,b,c | a8=1, b4=a4, c4=a2, bab-1=a-1, cac-1=a5, cbc-1=a3b >
Subgroups: 120 in 64 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4.Q8, C2.D8, C8.C4, C8.C4, C2×C16, M5(2), M5(2), C42⋊C2, C22×C8, C2×M4(2), C23.25D4, C2×C8.C4, C2×M5(2), C8.4C42
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, C8.4C42
►On 32 points
(1 28 5 32 9 20 13 24)(2 21 6 25 10 29 14 17)(3 30 7 18 11 22 15 26)(4 23 8 27 12 31 16 19)
(1 16 30 29 9 8 22 21)(2 28 31 7 10 20 23 15)(3 14 32 27 11 6 24 19)(4 26 17 5 12 18 25 13)
(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)| (1,28,5,32,9,20,13,24)(2,21,6,25,10,29,14,17)(3,30,7,18,11,22,15,26)(4,23,8,27,12,31,16,19), (1,16,30,29,9,8,22,21)(2,28,31,7,10,20,23,15)(3,14,32,27,11,6,24,19)(4,26,17,5,12,18,25,13), (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( (1,28,5,32,9,20,13,24)(2,21,6,25,10,29,14,17)(3,30,7,18,11,22,15,26)(4,23,8,27,12,31,16,19), (1,16,30,29,9,8,22,21)(2,28,31,7,10,20,23,15)(3,14,32,27,11,6,24,19)(4,26,17,5,12,18,25,13), (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([[(1,28,5,32,9,20,13,24),(2,21,6,25,10,29,14,17),(3,30,7,18,11,22,15,26),(4,23,8,27,12,31,16,19)], [(1,16,30,29,9,8,22,21),(2,28,31,7,10,20,23,15),(3,14,32,27,11,6,24,19),(4,26,17,5,12,18,25,13)], [(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 | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | SD16 | C8.4C42 |
| kernel | C8.4C42 | C23.25D4 | C2×C8.C4 | C2×M5(2) | C4.Q8 | C8.C4 | M5(2) | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of C8.4C42 ►in GL4(𝔽17) generated by
| 12 | 12 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 13 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,5,12,0,0,5,5],[0,0,0,1,0,0,1,0,0,13,0,0,13,0,0,0],[0,0,14,14,0,0,3,14,1,0,0,0,0,1,0,0] >;
C8.4C42 in GAP, Magma, Sage, TeX
C_8._4C_4^2
% in TeX
G:=Group("C8.4C4^2"); // GroupNames label
G:=SmallGroup(128,121);
// by ID
G=gap.SmallGroup(128,121);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,248,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^4,c^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;
// generators/relations