p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.398D4, C42.599C23, D4⋊C8⋊2C2, (C2×D4)⋊6C8, (C4×C8)⋊1C22, D4.5(C2×C8), C4⋊C8⋊55C22, (C4×D4).14C4, C4.5(C22×C8), C42.52(C2×C4), C4.5(C2×M4(2)), C4.11(C22⋊C8), (C22×D4).23C4, (C22×C4).655D4, (C2×C4).16M4(2), C4.130(C8⋊C22), C42.12C4⋊7C2, (C4×D4).263C22, C22.27(C22⋊C8), (C2×C42).155C22, C23.167(C22⋊C4), C2.2(C42⋊C22), C2.1(C23.37D4), (C2×C4×D4).6C2, (C2×C4⋊C4).38C4, (C2×C4).17(C2×C8), C4⋊C4.177(C2×C4), C2.14(C2×C22⋊C8), (C2×D4).191(C2×C4), (C2×C4).1442(C2×D4), (C2×C4).304(C22×C4), (C22×C4).177(C2×C4), C22.98(C2×C22⋊C4), (C2×C4).237(C22⋊C4), SmallGroup(128,210)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.398D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 332 in 156 conjugacy classes, 62 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, D4⋊C8, C42.12C4, C2×C4×D4, C42.398D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C8⋊C22, C2×C22⋊C8, C23.37D4, C42⋊C22, C42.398D4
►On 32 points
(1 9 27 23)(2 24 28 10)(3 11 29 17)(4 18 30 12)(5 13 31 19)(6 20 32 14)(7 15 25 21)(8 22 26 16)
(1 21 31 11)(2 22 32 12)(3 23 25 13)(4 24 26 14)(5 17 27 15)(6 18 28 16)(7 19 29 9)(8 20 30 10)
(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 20 21 30 31 10 11 8)(2 29 22 9 32 7 12 19)(3 16 23 6 25 18 13 28)(4 5 24 17 26 27 14 15)
G:=sub<Sym(32)| (1,9,27,23)(2,24,28,10)(3,11,29,17)(4,18,30,12)(5,13,31,19)(6,20,32,14)(7,15,25,21)(8,22,26,16), (1,21,31,11)(2,22,32,12)(3,23,25,13)(4,24,26,14)(5,17,27,15)(6,18,28,16)(7,19,29,9)(8,20,30,10), (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), (1,20,21,30,31,10,11,8)(2,29,22,9,32,7,12,19)(3,16,23,6,25,18,13,28)(4,5,24,17,26,27,14,15)>;
G:=Group( (1,9,27,23)(2,24,28,10)(3,11,29,17)(4,18,30,12)(5,13,31,19)(6,20,32,14)(7,15,25,21)(8,22,26,16), (1,21,31,11)(2,22,32,12)(3,23,25,13)(4,24,26,14)(5,17,27,15)(6,18,28,16)(7,19,29,9)(8,20,30,10), (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), (1,20,21,30,31,10,11,8)(2,29,22,9,32,7,12,19)(3,16,23,6,25,18,13,28)(4,5,24,17,26,27,14,15) );
G=PermutationGroup([[(1,9,27,23),(2,24,28,10),(3,11,29,17),(4,18,30,12),(5,13,31,19),(6,20,32,14),(7,15,25,21),(8,22,26,16)], [(1,21,31,11),(2,22,32,12),(3,23,25,13),(4,24,26,14),(5,17,27,15),(6,18,28,16),(7,19,29,9),(8,20,30,10)], [(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)], [(1,20,21,30,31,10,11,8),(2,29,22,9,32,7,12,19),(3,16,23,6,25,18,13,28),(4,5,24,17,26,27,14,15)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 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 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D4 | M4(2) | C8⋊C22 | C42⋊C22 |
| kernel | C42.398D4 | D4⋊C8 | C42.12C4 | C2×C4×D4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C2×D4 | C42 | C22×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 16 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C42.398D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 6 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 4 | 15 | 0 | 0 | 0 | 0 |
| 10 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,6,0,0,0,0,15,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,1,1,0,0,0,0,0,16,0,0],[4,10,0,0,0,0,15,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.398D4 in GAP, Magma, Sage, TeX
C_4^2._{398}D_4 % in TeX
G:=Group("C4^2.398D4"); // GroupNames label
G:=SmallGroup(128,210);
// by ID
G=gap.SmallGroup(128,210);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,184,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations