p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊2D4, C23.17D4, (C2×D8)⋊7C2, C4.Q8⋊4C2, C4⋊D4⋊4C2, C4.57(C2×D4), (C2×C4).31D4, C4⋊C4.9C22, D4⋊C4⋊18C2, C4.11(C4○D4), (C2×M4(2))⋊2C2, (C2×C4).97C23, (C2×C8).16C22, C22.93(C2×D4), C2.21(C4⋊D4), C2.13(C8⋊C22), (C2×D4).18C22, (C22×C4).49C22, SmallGroup(64,150)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊2D4
G = < a,b,c | a8=b4=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >
Subgroups: 141 in 65 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C8⋊2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C8⋊C22, C8⋊2D4
Character table of C8⋊2D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 4 | 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 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 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 | orthogonal lifted from C8⋊C22 |
►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 11 27 19)(2 14 28 22)(3 9 29 17)(4 12 30 20)(5 15 31 23)(6 10 32 18)(7 13 25 21)(8 16 26 24)
(1 8)(2 7)(3 6)(4 5)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 28)(26 27)(29 32)(30 31)
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), (1,11,27,19)(2,14,28,22)(3,9,29,17)(4,12,30,20)(5,15,31,23)(6,10,32,18)(7,13,25,21)(8,16,26,24), (1,8)(2,7)(3,6)(4,5)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,28)(26,27)(29,32)(30,31)>;
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), (1,11,27,19)(2,14,28,22)(3,9,29,17)(4,12,30,20)(5,15,31,23)(6,10,32,18)(7,13,25,21)(8,16,26,24), (1,8)(2,7)(3,6)(4,5)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,28)(26,27)(29,32)(30,31) );
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)], [(1,11,27,19),(2,14,28,22),(3,9,29,17),(4,12,30,20),(5,15,31,23),(6,10,32,18),(7,13,25,21),(8,16,26,24)], [(1,8),(2,7),(3,6),(4,5),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(25,28),(26,27),(29,32),(30,31)]])
C8⋊2D4 is a maximal subgroup of
C23.SD16 M5(2).C22 C24.110D4 M4(2)⋊14D4 (C2×C8)⋊12D4 (C2×C8)⋊13D4 M4(2)⋊16D4 C42.260D4 C42.261D4 C24.125D4 C24.127D4 C24.130D4 C4.2+ 1+4 C4.142+ 1+4 C4.192+ 1+4 C42.299D4 C42.301D4 C42.304D4
C23.D4p: C23.D8 D8⋊D4 C24⋊3D4 C40⋊3D4 C56⋊3D4 ...
(C2p×D8)⋊C2: C42.255D4 C42.388C23 C42.391C23 C42.471C23 C42.474C23 C42.495C23 C42.496C23 C24⋊12D4 ...
C4⋊C4.D2p: C42.385C23 C4.2- 1+4 C42.26C23 C42.30C23 D8⋊9D4 Q16⋊10D4 SD16⋊1D4 SD16⋊2D4 ...
C8⋊2D4 is a maximal quotient of
C24.76D4 C23⋊2D8
C8⋊D4p: C8⋊2D8 C24⋊7D4 C8⋊2D20 C56⋊7D4 ...
C4⋊C4.D2p: C8⋊2SD16 C8⋊4SD16 C8⋊2Q16 C42.252C23 C42.253C23 C24.84D4 C4⋊C4.106D4 C3⋊C8⋊D4 ...
(C2×C8).D2p: C24.67D4 C2.(C8⋊2D4) C8⋊(C4⋊C4) (C2×D8)⋊10C4 (C2×C4)⋊3D8 C24.88D4 (C2×C4).21Q16 C24⋊3D4 ...
Matrix representation of C8⋊2D4 ►in GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 11 | 14 | 3 | 14 |
| 0 | 0 | 3 | 3 | 3 | 3 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 11 | 6 |
| 0 | 0 | 14 | 0 | 0 | 11 |
| 0 | 0 | 6 | 3 | 14 | 3 |
| 0 | 0 | 14 | 3 | 3 | 14 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 11 | 14 | 3 | 14 |
| 0 | 0 | 3 | 3 | 14 | 14 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,11,3,11,3,0,0,11,0,14,3,0,0,0,0,3,3,0,0,0,0,14,3],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,6,14,6,14,0,0,0,0,3,3,0,0,11,0,14,3,0,0,6,11,3,14],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,11,3,11,3,0,0,11,6,14,3,0,0,0,0,3,14,0,0,0,0,14,14] >;
C8⋊2D4 in GAP, Magma, Sage, TeX
C_8\rtimes_2D_4
% in TeX
G:=Group("C8:2D4"); // GroupNames label
G:=SmallGroup(64,150);
// by ID
G=gap.SmallGroup(64,150);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,332,963,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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