p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.372D4, C24.4(C2×C4), C23⋊C8.10C2, C42.12C4⋊4C2, C42.6C4⋊24C2, C22.14(C8○D4), C22⋊C8.159C22, (C22×C4).437C23, C23.174(C22×C4), (C2×C42).153C22, C22.M4(2)⋊17C2, C2.11(C23.C23), C2.9(M4(2).8C22), (C2×C4⋊C4).15C4, (C2×C4).1134(C2×D4), (C2×C4⋊C4).12C22, (C2×C22⋊C4).10C4, (C22×C4).14(C2×C4), (C2×C4).73(C22⋊C4), (C2×C42⋊2C2).1C2, (C2×C22⋊C4).89C22, C22.155(C2×C22⋊C4), C2.11((C22×C8)⋊C2), SmallGroup(128,205)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.372D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=ab2, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 220 in 107 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C23⋊C8, C22.M4(2), C42.12C4, C42.6C4, C2×C42⋊2C2, C42.372D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8○D4, (C22×C8)⋊C2, C23.C23, M4(2).8C22, C42.372D4
►On 32 points
(1 15 31 19)(2 12 32 24)(3 9 25 21)(4 14 26 18)(5 11 27 23)(6 16 28 20)(7 13 29 17)(8 10 30 22)
(1 3 5 7)(2 30 6 26)(4 32 8 28)(9 11 13 15)(10 20 14 24)(12 22 16 18)(17 19 21 23)(25 27 29 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)
(1 28 3 4 5 32 7 8)(2 29 30 31 6 25 26 27)(9 14 11 24 13 10 15 20)(12 17 22 19 16 21 18 23)
G:=sub<Sym(32)| (1,15,31,19)(2,12,32,24)(3,9,25,21)(4,14,26,18)(5,11,27,23)(6,16,28,20)(7,13,29,17)(8,10,30,22), (1,3,5,7)(2,30,6,26)(4,32,8,28)(9,11,13,15)(10,20,14,24)(12,22,16,18)(17,19,21,23)(25,27,29,31), (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), (1,28,3,4,5,32,7,8)(2,29,30,31,6,25,26,27)(9,14,11,24,13,10,15,20)(12,17,22,19,16,21,18,23)>;
G:=Group( (1,15,31,19)(2,12,32,24)(3,9,25,21)(4,14,26,18)(5,11,27,23)(6,16,28,20)(7,13,29,17)(8,10,30,22), (1,3,5,7)(2,30,6,26)(4,32,8,28)(9,11,13,15)(10,20,14,24)(12,22,16,18)(17,19,21,23)(25,27,29,31), (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), (1,28,3,4,5,32,7,8)(2,29,30,31,6,25,26,27)(9,14,11,24,13,10,15,20)(12,17,22,19,16,21,18,23) );
G=PermutationGroup([[(1,15,31,19),(2,12,32,24),(3,9,25,21),(4,14,26,18),(5,11,27,23),(6,16,28,20),(7,13,29,17),(8,10,30,22)], [(1,3,5,7),(2,30,6,26),(4,32,8,28),(9,11,13,15),(10,20,14,24),(12,22,16,18),(17,19,21,23),(25,27,29,31)], [(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)], [(1,28,3,4,5,32,7,8),(2,29,30,31,6,25,26,27),(9,14,11,24,13,10,15,20),(12,17,22,19,16,21,18,23)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | C8○D4 | C23.C23 | M4(2).8C22 |
| kernel | C42.372D4 | C23⋊C8 | C22.M4(2) | C42.12C4 | C42.6C4 | C2×C42⋊2C2 | C2×C22⋊C4 | C2×C4⋊C4 | C42 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C42.372D4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 13 |
| 0 | 0 | 0 | 15 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 1 |
| 0 | 0 | 0 | 9 | 16 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 2 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 15 | 0 |
| 0 | 0 | 8 | 0 | 13 | 16 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 16 | 0 | 9 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 15 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,8,16,8,15,0,0,0,0,0,4,0,0,0,0,13,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,15,13,15,9,0,0,0,0,0,16,0,0,0,0,1,0],[2,2,0,0,0,0,13,15,0,0,0,0,0,0,2,8,0,16,0,0,0,0,1,0,0,0,15,13,15,9,0,0,0,16,0,0],[15,15,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,15,13,15,9,0,0,0,1,0,0] >;
C42.372D4 in GAP, Magma, Sage, TeX
C_4^2._{372}D_4 % in TeX
G:=Group("C4^2.372D4"); // GroupNames label
G:=SmallGroup(128,205);
// by ID
G=gap.SmallGroup(128,205);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,520,1123,851,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations