p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊1D4, C23.16D4, C2.D8⋊12C2, (C2×C4).30D4, C4.56(C2×D4), C22⋊Q8⋊4C2, (C2×SD16)⋊1C2, C4⋊D4.4C2, C4⋊C4.8C22, D4⋊C4⋊17C2, Q8⋊C4⋊17C2, C4.10(C4○D4), (C2×M4(2))⋊1C2, (C2×C4).96C23, (C2×C8).52C22, C22.92(C2×D4), C2.20(C4⋊D4), C2.12(C8⋊C22), (C2×D4).17C22, (C2×Q8).13C22, C2.12(C8.C22), (C22×C4).48C22, SmallGroup(64,149)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊D4
G = < a,b,c | a8=b4=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >
Subgroups: 117 in 60 conjugacy classes, 27 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C8⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C8⋊C22, C8.C22, C8⋊D4
Character table of C8⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ16 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 32 points
(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 23 27 12)(2 22 28 11)(3 21 29 10)(4 20 30 9)(5 19 31 16)(6 18 32 15)(7 17 25 14)(8 24 26 13)
(2 4)(3 7)(6 8)(9 22)(10 17)(11 20)(12 23)(13 18)(14 21)(15 24)(16 19)(25 29)(26 32)(28 30)
G:=sub<Sym(32)| (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), (1,23,27,12)(2,22,28,11)(3,21,29,10)(4,20,30,9)(5,19,31,16)(6,18,32,15)(7,17,25,14)(8,24,26,13), (2,4)(3,7)(6,8)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19)(25,29)(26,32)(28,30)>;
G:=Group( (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), (1,23,27,12)(2,22,28,11)(3,21,29,10)(4,20,30,9)(5,19,31,16)(6,18,32,15)(7,17,25,14)(8,24,26,13), (2,4)(3,7)(6,8)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19)(25,29)(26,32)(28,30) );
G=PermutationGroup([[(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)], [(1,23,27,12),(2,22,28,11),(3,21,29,10),(4,20,30,9),(5,19,31,16),(6,18,32,15),(7,17,25,14),(8,24,26,13)], [(2,4),(3,7),(6,8),(9,22),(10,17),(11,20),(12,23),(13,18),(14,21),(15,24),(16,19),(25,29),(26,32),(28,30)]])
C8⋊D4 is a maximal subgroup of
C42.256D4 C42.260D4 C24.126D4 C24.129D4 C4.152+ 1+4 C4.162+ 1+4 C42.300D4 C42.302D4 C42.304D4
C4⋊C4.D2p: C42.386C23 C42.387C23 C42.390C23 C42.391C23 C4.2- 1+4 C42.25C23 C42.27C23 C42.29C23 ...
C8⋊pD4⋊C2: C24.110D4 M4(2)⋊14D4 (C2×C8)⋊14D4 M4(2)⋊16D4 C42.255D4 C42.259D4 C24.127D4 C24.130D4 ...
C4p.(C2×D4): M4(2)⋊15D4 (C2×C8)⋊11D4 M4(2)⋊17D4 C24⋊2D4 C24⋊8D4 C40⋊2D4 C40⋊8D4 C56⋊2D4 ...
C8⋊D4 is a maximal quotient of
C24.76D4 C23⋊3SD16
C8⋊D4p: C8⋊D8 C8⋊3D12 C8⋊3D20 C8⋊3D28 ...
C23.D4p: C24.83D4 C24⋊2D4 C40⋊2D4 C56⋊2D4 ...
C4⋊C4.D2p: C8⋊SD16 C8⋊3SD16 C8⋊Q16 C42.248C23 C42.249C23 C42.250C23 C42.251C23 C24.86D4 ...
(C2×C8).D2p: C24.67D4 C24.75D4 C2.(C8⋊D4) C2.(C8⋊2D4) C4.(C4×Q8) C8⋊(C22⋊C4) (C2×C4)⋊5SD16 C24.89D4 ...
Matrix representation of C8⋊D4 ►in GL6(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | -1 | -1 | -1 | -2 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | -1 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | -1 | -1 | -2 |
| 0 | 0 | 0 | 1 | 0 | 1 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,1,-1,0,0,0,0,0,-2,0,1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,1,-1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,2,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,-1,-1,1,0,0,0,0,-1,0,0,0,0,0,-2,1] >;
C8⋊D4 in GAP, Magma, Sage, TeX
C_8\rtimes D_4
% in TeX
G:=Group("C8:D4"); // GroupNames label
G:=SmallGroup(64,149);
// by ID
G=gap.SmallGroup(64,149);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,332,963,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations
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