p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊2M4(2), C42.11C4, C42.67C22, C4⋊C8⋊11C2, (C2×C4).71D4, C4.72(C2×D4), C4.11(C4⋊C4), (C2×C4).16Q8, C4.21(C2×Q8), C22.7(C4⋊C4), (C22×C4).15C4, C23.31(C2×C4), (C2×C42).14C2, (C2×C8).45C22, C2.7(C2×M4(2)), (C2×C4).148C23, (C2×M4(2)).13C2, C22.43(C22×C4), (C22×C4).111C22, C2.9(C2×C4⋊C4), (C2×C4).56(C2×C4), SmallGroup(64,104)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊M4(2)
G = < a,b,c | a4=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >
Subgroups: 81 in 63 conjugacy classes, 45 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C4⋊C8, C2×C42, C2×M4(2), C4⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C2×M4(2), C4⋊M4(2)
Character table of C4⋊M4(2)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | i | -i | i | -i | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | -2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | -2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | -2 | 2 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ28 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | -2 | 2 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
►On 32 points
(1 24 31 16)(2 9 32 17)(3 18 25 10)(4 11 26 19)(5 20 27 12)(6 13 28 21)(7 22 29 14)(8 15 30 23)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
G:=sub<Sym(32)| (1,24,31,16)(2,9,32,17)(3,18,25,10)(4,11,26,19)(5,20,27,12)(6,13,28,21)(7,22,29,14)(8,15,30,23), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)>;
G:=Group( (1,24,31,16)(2,9,32,17)(3,18,25,10)(4,11,26,19)(5,20,27,12)(6,13,28,21)(7,22,29,14)(8,15,30,23), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32) );
G=PermutationGroup([[(1,24,31,16),(2,9,32,17),(3,18,25,10),(4,11,26,19),(5,20,27,12),(6,13,28,21),(7,22,29,14),(8,15,30,23)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)]])
C4⋊M4(2) is a maximal subgroup of
C42.5Q8 C42.27D4 C42.388D4 C42.389D4 C42.10Q8 C42.30D4 C42.32D4 C42.C8 D4⋊M4(2) Q8⋊M4(2) D4⋊4M4(2) C42.315D4 C42.316D4 C42.52D4 C42.413D4 C42.414D4 C42.78D4 C42.415D4 C42.416D4 C42.79D4 C42.84D4 C42.86D4 C42.87D4 C42.88D4 C42.90D4 C42.91D4 C42.21Q8 C42.96D4 C42.97D4 C42.102D4 C42.322D4 C42.104D4 M4(2).41D4 C42.430D4 C43⋊C2 C42⋊8D4 M4(2)⋊12D4 C42.114D4 C42.115D4 C42⋊16Q8 C42⋊Q8 M4(2)⋊8Q8 C42.128D4 C42.131D4 C42.32Q8 C42.257C23 C42.674C23 C42.677C23 C42.259C23 C42.678C23 C42.681C23 C42.266C23 D4×M4(2) C42.286C23 C42.287C23 Q8×M4(2) C42.698C23 D4⋊8M4(2) Q8⋊7M4(2) C42.308C23 C42.310C23 C42.211D4 C42.212D4 C42.444D4 C42.445D4 C42.446D4 C42.219D4 C42.220D4 C42.448D4 C42.449D4 C42.299D4 C42.300D4 C42.301D4 C42.302D4 C42.303D4 C42.304D4
C4p⋊M4(2): C8⋊8M4(2) C8⋊7M4(2) C8⋊1M4(2) C12⋊M4(2) C12⋊7M4(2) C20⋊5M4(2) C20⋊13M4(2) C20⋊3M4(2) ...
C4p.(C4⋊C4): C42.324D4 C42.106D4 Dic3⋊4M4(2) Dic5⋊5M4(2) C42.14F5 Dic7⋊4M4(2) ...
C4⋊M4(2) is a maximal quotient of
C42.43Q8 C23.28C42 C43.7C2 C42.425D4 C42⋊8C8 C42⋊9C8 (C2×C8).195D4 C42.14F5
C4p⋊M4(2): C8⋊8M4(2) C8⋊7M4(2) C8⋊1M4(2) C12⋊M4(2) C12⋊7M4(2) C20⋊5M4(2) C20⋊13M4(2) C20⋊3M4(2) ...
(C2×C4p).Q8: C42.27Q8 Dic3⋊4M4(2) Dic5⋊5M4(2) Dic7⋊4M4(2) ...
Matrix representation of C4⋊M4(2) ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C4⋊M4(2) in GAP, Magma, Sage, TeX
C_4\rtimes M_4(2)
% in TeX
G:=Group("C4:M4(2)"); // GroupNames label
G:=SmallGroup(64,104);
// by ID
G=gap.SmallGroup(64,104);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,650,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations
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