p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.55D4, C4⋊Q8⋊8C4, (C4×D4)⋊3C4, C4.13C4≀C2, C4⋊1D4⋊7C4, C42⋊4C4⋊5C2, C42.72(C2×C4), (C22×C4).734D4, C23.495(C2×D4), C42.6C4⋊29C2, C22.SD16⋊18C2, C4⋊D4.133C22, C22⋊C8.128C22, C22.34(C8⋊C22), (C22×C4).627C23, (C2×C42).175C22, C2.7(C23.37D4), C22.26C24.6C2, C2.C42.504C22, C2.15(C23.C23), C4⋊C4.5(C2×C4), C2.22(C2×C4≀C2), (C2×D4).6(C2×C4), (C2×C4).1151(C2×D4), (C2×C4).117(C22×C4), (C2×C4).317(C22⋊C4), C22.181(C2×C22⋊C4), SmallGroup(128,237)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.55D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >
Subgroups: 316 in 136 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, 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, C2.C42, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C22.SD16, C42⋊4C4, C42.6C4, C22.26C24, C42.55D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C8⋊C22, C23.C23, C23.37D4, C2×C4≀C2, C42.55D4
►On 32 points
(1 10 31 21)(2 22 32 11)(3 12 25 23)(4 24 26 13)(5 14 27 17)(6 18 28 15)(7 16 29 19)(8 20 30 9)
(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)
(1 17 31 14)(2 20 6 24)(3 16 25 19)(4 11 8 15)(5 21 27 10)(7 12 29 23)(9 28 13 32)(18 26 22 30)
(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)| (1,10,31,21)(2,22,32,11)(3,12,25,23)(4,24,26,13)(5,14,27,17)(6,18,28,15)(7,16,29,19)(8,20,30,9), (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), (1,17,31,14)(2,20,6,24)(3,16,25,19)(4,11,8,15)(5,21,27,10)(7,12,29,23)(9,28,13,32)(18,26,22,30), (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( (1,10,31,21)(2,22,32,11)(3,12,25,23)(4,24,26,13)(5,14,27,17)(6,18,28,15)(7,16,29,19)(8,20,30,9), (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), (1,17,31,14)(2,20,6,24)(3,16,25,19)(4,11,8,15)(5,21,27,10)(7,12,29,23)(9,28,13,32)(18,26,22,30), (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([[(1,10,31,21),(2,22,32,11),(3,12,25,23),(4,24,26,13),(5,14,27,17),(6,18,28,15),(7,16,29,19),(8,20,30,9)], [(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)], [(1,17,31,14),(2,20,6,24),(3,16,25,19),(4,11,8,15),(5,21,27,10),(7,12,29,23),(9,28,13,32),(18,26,22,30)], [(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 | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C8⋊C22 | C23.C23 |
| kernel | C42.55D4 | C22.SD16 | C42⋊4C4 | C42.6C4 | C22.26C24 | C4×D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.55D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 2 | 5 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 8 | 3 |
| 0 | 0 | 2 | 1 | 9 | 14 |
| 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 16 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 8 | 16 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 4 | 11 | 1 | 11 |
| 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,2,0,13,0,0,0,5,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,2,0,0,0,0,0,1,0,0,0,0,8,9,13,2,0,0,3,14,1,4],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,13,4,0,0,0,0,0,0,16,9,0,0,16,0,0,1],[0,1,0,0,0,0,13,0,0,0,0,0,0,0,16,0,4,0,0,0,16,0,11,4,0,0,8,9,1,0,0,0,16,1,11,0] >;
C42.55D4 in GAP, Magma, Sage, TeX
C_4^2._{55}D_4 % in TeX
G:=Group("C4^2.55D4"); // GroupNames label
G:=SmallGroup(128,237);
// by ID
G=gap.SmallGroup(128,237);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,184,1123,1018,248,1971]);
// 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,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations