p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.403D4, C4⋊Q8⋊6C4, (C4×D4)⋊2C4, C4.24C4≀C2, C4⋊1D4⋊6C4, (C2×C4).128D8, C42.70(C2×C4), C22.10(C2×D8), (C2×C4).115SD16, (C22×C4).732D4, C23.492(C2×D4), C4.54(D4⋊C4), C4○3(C22.SD16), C22.SD16⋊24C2, C22.28(C2×SD16), C42.12C4⋊16C2, C4⋊D4.131C22, C22⋊C8.163C22, (C2×C42).173C22, (C22×C4).624C23, C22.26C24.5C2, C2.C42.501C22, C2.13(C23.C23), (C4×C4⋊C4)⋊1C2, C4⋊C4.3(C2×C4), C2.19(C2×C4≀C2), (C2×D4).5(C2×C4), C2.8(C2×D4⋊C4), (C2×C4).1148(C2×D4), (C2×C4).87(C22⋊C4), (C2×C4).114(C22×C4), C22.178(C2×C22⋊C4), SmallGroup(128,234)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.403D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 324 in 142 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C22.SD16, C4×C4⋊C4, C42.12C4, C22.26C24, C42.403D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C4≀C2, C2×C22⋊C4, C2×D8, C2×SD16, C23.C23, C2×D4⋊C4, C2×C4≀C2, C42.403D4
►On 32 points
(1 15 25 18)(2 16 26 19)(3 9 27 20)(4 10 28 21)(5 11 29 22)(6 12 30 23)(7 13 31 24)(8 14 32 17)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(2 32 26 8)(3 31)(4 6 28 30)(7 27)(9 24)(10 12 21 23)(13 20)(14 16 17 19)
(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)| (1,15,25,18)(2,16,26,19)(3,9,27,20)(4,10,28,21)(5,11,29,22)(6,12,30,23)(7,13,31,24)(8,14,32,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,32,26,8)(3,31)(4,6,28,30)(7,27)(9,24)(10,12,21,23)(13,20)(14,16,17,19), (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( (1,15,25,18)(2,16,26,19)(3,9,27,20)(4,10,28,21)(5,11,29,22)(6,12,30,23)(7,13,31,24)(8,14,32,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,32,26,8)(3,31)(4,6,28,30)(7,27)(9,24)(10,12,21,23)(13,20)(14,16,17,19), (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([[(1,15,25,18),(2,16,26,19),(3,9,27,20),(4,10,28,21),(5,11,29,22),(6,12,30,23),(7,13,31,24),(8,14,32,17)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(2,32,26,8),(3,31),(4,6,28,30),(7,27),(9,24),(10,12,21,23),(13,20),(14,16,17,19)], [(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 | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | C4≀C2 | C23.C23 |
| kernel | C42.403D4 | C22.SD16 | C4×C4⋊C4 | C42.12C4 | C22.26C24 | C4×D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.403D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,4,0,0,0,0,1,1,0,0,0,16],[0,1,0,0,4,0,0,0,0,0,0,3,0,0,11,11] >;
C42.403D4 in GAP, Magma, Sage, TeX
C_4^2._{403}D_4 % in TeX
G:=Group("C4^2.403D4"); // GroupNames label
G:=SmallGroup(128,234);
// by ID
G=gap.SmallGroup(128,234);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,520,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations