p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.413D4, C42.163C23, C4.55(C2×D8), (C2×C4).17D8, C4⋊C8⋊46C22, C4⋊1D4.14C4, C4.D8⋊20C2, C4.75(C2×SD16), (C2×C4).26SD16, C42.104(C2×C4), C4.6(C4.D4), (C22×D4).11C4, (C22×C4).236D4, C4.20(D4⋊C4), C4.108(C8⋊C22), C4⋊M4(2)⋊19C2, C42.12C4⋊21C2, C4⋊1D4.126C22, (C2×C42).207C22, C22.26(D4⋊C4), C23.181(C22⋊C4), C2.11(C23.37D4), (C2×D4).30(C2×C4), (C2×C4⋊1D4).2C2, C2.14(C2×D4⋊C4), (C2×C4).1234(C2×D4), C2.18(C2×C4.D4), (C22×C4).229(C2×C4), (C2×C4).157(C22×C4), (C2×C4).245(C22⋊C4), C22.221(C2×C22⋊C4), SmallGroup(128,277)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.413D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 492 in 172 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4⋊1D4, C4⋊1D4, C2×M4(2), C22×D4, C22×D4, C4.D8, C4⋊M4(2), C42.12C4, C2×C4⋊1D4, C42.413D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4.D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C8⋊C22, C2×C4.D4, C2×D4⋊C4, C23.37D4, C42.413D4
►On 32 points
(1 32 18 9)(2 10 19 25)(3 26 20 11)(4 12 21 27)(5 28 22 13)(6 14 23 29)(7 30 24 15)(8 16 17 31)
(1 11 22 30)(2 31 23 12)(3 13 24 32)(4 25 17 14)(5 15 18 26)(6 27 19 16)(7 9 20 28)(8 29 21 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 21 11 10 22 8 30 29)(2 28 31 7 23 9 12 20)(3 19 13 16 24 6 32 27)(4 26 25 5 17 15 14 18)
G:=sub<Sym(32)| (1,32,18,9)(2,10,19,25)(3,26,20,11)(4,12,21,27)(5,28,22,13)(6,14,23,29)(7,30,24,15)(8,16,17,31), (1,11,22,30)(2,31,23,12)(3,13,24,32)(4,25,17,14)(5,15,18,26)(6,27,19,16)(7,9,20,28)(8,29,21,10), (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), (1,21,11,10,22,8,30,29)(2,28,31,7,23,9,12,20)(3,19,13,16,24,6,32,27)(4,26,25,5,17,15,14,18)>;
G:=Group( (1,32,18,9)(2,10,19,25)(3,26,20,11)(4,12,21,27)(5,28,22,13)(6,14,23,29)(7,30,24,15)(8,16,17,31), (1,11,22,30)(2,31,23,12)(3,13,24,32)(4,25,17,14)(5,15,18,26)(6,27,19,16)(7,9,20,28)(8,29,21,10), (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), (1,21,11,10,22,8,30,29)(2,28,31,7,23,9,12,20)(3,19,13,16,24,6,32,27)(4,26,25,5,17,15,14,18) );
G=PermutationGroup([[(1,32,18,9),(2,10,19,25),(3,26,20,11),(4,12,21,27),(5,28,22,13),(6,14,23,29),(7,30,24,15),(8,16,17,31)], [(1,11,22,30),(2,31,23,12),(3,13,24,32),(4,25,17,14),(5,15,18,26),(6,27,19,16),(7,9,20,28),(8,29,21,10)], [(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)], [(1,21,11,10,22,8,30,29),(2,28,31,7,23,9,12,20),(3,19,13,16,24,6,32,27),(4,26,25,5,17,15,14,18)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D8 | SD16 | C4.D4 | C8⋊C22 |
| kernel | C42.413D4 | C4.D8 | C4⋊M4(2) | C42.12C4 | C2×C4⋊1D4 | C4⋊1D4 | C22×D4 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.413D4 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 2 |
| 0 | 0 | 0 | 8 | 2 | 0 |
| 0 | 0 | 9 | 2 | 9 | 0 |
| 0 | 0 | 2 | 8 | 0 | 8 |
| 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 2 | 0 |
| 0 | 0 | 9 | 0 | 0 | 2 |
| 0 | 0 | 15 | 9 | 0 | 9 |
| 0 | 0 | 8 | 15 | 8 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,9,0,9,2,0,0,0,8,2,8,0,0,0,2,9,0,0,0,2,0,0,8],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,0,9,15,8,0,0,8,0,9,15,0,0,2,0,0,8,0,0,0,2,9,0] >;
C42.413D4 in GAP, Magma, Sage, TeX
C_4^2._{413}D_4 % in TeX
G:=Group("C4^2.413D4"); // GroupNames label
G:=SmallGroup(128,277);
// by ID
G=gap.SmallGroup(128,277);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,1123,1018,248,1971,242]);
// 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,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations