p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.9Q8, C42.369D4, C4⋊C8⋊3C4, C4.38C4≀C2, C8⋊C4⋊11C4, C22.12C4≀C2, C42.38(C2×C4), (C2×C4).12C42, C42⋊4C4.4C2, (C22×C4).642D4, (C4×M4(2)).12C2, C42.6C4.9C2, C2.12(C42⋊6C4), C2.C42.12C4, (C2×C42).135C22, C2.8(M4(2)⋊4C4), C23.142(C22⋊C4), C22.50(C2.C42), (C2×C4).21(C4⋊C4), (C22×C4).155(C2×C4), (C2×C4).303(C22⋊C4), SmallGroup(128,32)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.9Q8
G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=ab2c-1 >
Subgroups: 160 in 86 conjugacy classes, 34 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×13], C23, C42 [×4], C42 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C2.C42, C4×C8, C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C2×C42, C2×M4(2), C42⋊4C4, C4×M4(2), C42.6C4, C42.9Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4≀C2 [×4], C42⋊6C4 [×2], M4(2)⋊4C4, C42.9Q8
►On 32 points
(1 17 31 14)(2 22 32 11)(3 19 25 16)(4 24 26 13)(5 21 27 10)(6 18 28 15)(7 23 29 12)(8 20 30 9)
(1 7 5 3)(2 26 6 30)(4 28 8 32)(9 22 13 18)(10 16 14 12)(11 24 15 20)(17 23 21 19)(25 31 29 27)
(1 7 5 3)(2 20 32 9)(4 22 26 11)(6 24 28 13)(8 18 30 15)(10 12 14 16)(17 19 21 23)(25 31 29 27)
(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)| (1,17,31,14)(2,22,32,11)(3,19,25,16)(4,24,26,13)(5,21,27,10)(6,18,28,15)(7,23,29,12)(8,20,30,9), (1,7,5,3)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(25,31,29,27), (1,7,5,3)(2,20,32,9)(4,22,26,11)(6,24,28,13)(8,18,30,15)(10,12,14,16)(17,19,21,23)(25,31,29,27), (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( (1,17,31,14)(2,22,32,11)(3,19,25,16)(4,24,26,13)(5,21,27,10)(6,18,28,15)(7,23,29,12)(8,20,30,9), (1,7,5,3)(2,26,6,30)(4,28,8,32)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(25,31,29,27), (1,7,5,3)(2,20,32,9)(4,22,26,11)(6,24,28,13)(8,18,30,15)(10,12,14,16)(17,19,21,23)(25,31,29,27), (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([(1,17,31,14),(2,22,32,11),(3,19,25,16),(4,24,26,13),(5,21,27,10),(6,18,28,15),(7,23,29,12),(8,20,30,9)], [(1,7,5,3),(2,26,6,30),(4,28,8,32),(9,22,13,18),(10,16,14,12),(11,24,15,20),(17,23,21,19),(25,31,29,27)], [(1,7,5,3),(2,20,32,9),(4,22,26,11),(6,24,28,13),(8,18,30,15),(10,12,14,16),(17,19,21,23),(25,31,29,27)], [(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)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | C4≀C2 | C4≀C2 | M4(2)⋊4C4 |
| kernel | C42.9Q8 | C42⋊4C4 | C4×M4(2) | C42.6C4 | C2.C42 | C8⋊C4 | C4⋊C8 | C42 | C42 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 8 | 8 | 2 |
Matrix representation of C42.9Q8 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 11 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,4],[6,11,0,0,11,11,0,0,0,0,0,16,0,0,1,0] >;
C42.9Q8 in GAP, Magma, Sage, TeX
C_4^2._9Q_8
% in TeX
G:=Group("C4^2.9Q8"); // GroupNames label
G:=SmallGroup(128,32);
// by ID
G=gap.SmallGroup(128,32);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*b^2*c^-1>;
// generators/relations