p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.11C42, C23.29SD16, (C2×C16)⋊5C4, C8.1(C4⋊C4), (C2×C8).2Q8, (C2×C8).81D4, (C2×C4).10D8, (C2×C4).6Q16, C4.Q8.3C4, C8.C4⋊2C4, C4.9(C2.D8), C2.2(C8.Q8), (C2×C4).93SD16, C8.41(C22⋊C4), C4.5(Q8⋊C4), (C22×C4).189D4, C4.31(D4⋊C4), (C2×M5(2)).11C2, C22.13(C4.Q8), C4.4(C2.C42), (C22×C8).202C22, C22.1(Q8⋊C4), C22.21(D4⋊C4), C2.13(C22.4Q16), (C2×C8).49(C2×C4), (C2×C4.Q8).1C2, (C2×C4).111(C4⋊C4), (C2×C8.C4).3C2, (C2×C4).62(C22⋊C4), SmallGroup(128,115)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.11C42
G = < a,b,c | a8=b4=1, c4=a2, bab-1=a3, ac=ca, cbc-1=a3b >
Subgroups: 136 in 68 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C16 [×2], C4⋊C4 [×3], C2×C8 [×6], C2×C8, M4(2) [×3], C22×C4, C22×C4, C4.Q8 [×2], C4.Q8, C8.C4 [×2], C8.C4, C2×C16 [×2], M5(2) [×2], C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4.Q8, C2×C8.C4, C2×M5(2), C8.11C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, C8.Q8 [×2], C8.11C42
►On 32 points
(1 19 5 23 9 27 13 31)(2 20 6 24 10 28 14 32)(3 21 7 25 11 29 15 17)(4 22 8 26 12 30 16 18)
(2 20 10 28)(3 7)(4 26 12 18)(5 13)(6 32 14 24)(8 22 16 30)(11 15)(19 23)(21 29)(27 31)
(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)| (1,19,5,23,9,27,13,31)(2,20,6,24,10,28,14,32)(3,21,7,25,11,29,15,17)(4,22,8,26,12,30,16,18), (2,20,10,28)(3,7)(4,26,12,18)(5,13)(6,32,14,24)(8,22,16,30)(11,15)(19,23)(21,29)(27,31), (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( (1,19,5,23,9,27,13,31)(2,20,6,24,10,28,14,32)(3,21,7,25,11,29,15,17)(4,22,8,26,12,30,16,18), (2,20,10,28)(3,7)(4,26,12,18)(5,13)(6,32,14,24)(8,22,16,30)(11,15)(19,23)(21,29)(27,31), (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([(1,19,5,23,9,27,13,31),(2,20,6,24,10,28,14,32),(3,21,7,25,11,29,15,17),(4,22,8,26,12,30,16,18)], [(2,20,10,28),(3,7),(4,26,12,18),(5,13),(6,32,14,24),(8,22,16,30),(11,15),(19,23),(21,29),(27,31)], [(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 | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 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.Q8 |
| kernel | C8.11C42 | C2×C4.Q8 | C2×C8.C4 | C2×M5(2) | C4.Q8 | C8.C4 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of C8.11C42 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 16 | 5 | 12 |
| 0 | 0 | 16 | 12 | 5 | 5 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 6 | 12 | 12 |
| 0 | 0 | 12 | 9 | 12 | 5 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 15 | 0 |
| 0 | 0 | 11 | 10 | 0 | 15 |
| 0 | 0 | 15 | 11 | 10 | 9 |
| 0 | 0 | 6 | 15 | 6 | 7 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,5,5,16,0,0,12,5,16,12,0,0,0,0,5,5,0,0,0,0,12,5],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,0,1,15,12,0,0,1,0,6,9,0,0,0,0,12,12,0,0,0,0,12,5],[1,0,0,0,0,0,8,16,0,0,0,0,0,0,7,11,15,6,0,0,8,10,11,15,0,0,15,0,10,6,0,0,0,15,9,7] >;
C8.11C42 in GAP, Magma, Sage, TeX
C_8._{11}C_4^2 % in TeX
G:=Group("C8.11C4^2"); // GroupNames label
G:=SmallGroup(128,115);
// by ID
G=gap.SmallGroup(128,115);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,1684,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^4=a^2,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations