direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4⋊D4, C23⋊4D4, C23.5C23, C24.12C22, C22.16C24, C4⋊3(C2×D4), (C2×C4)⋊10D4, C4⋊C4⋊9C22, (C23×C4)⋊5C2, C22⋊1(C2×D4), (C22×D4)⋊4C2, C2.5(C22×D4), (C2×D4)⋊10C22, (C2×C4).52C23, C22⋊C4⋊13C22, (C22×C4)⋊18C22, C22.29(C4○D4), (C2×C4⋊C4)⋊14C2, C2.5(C2×C4○D4), (C2×C22⋊C4)⋊9C2, SmallGroup(64,203)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊D4
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 353 in 213 conjugacy classes, 97 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C2×C4⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4
Character table of C2×C4⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 31)(14 32)(15 29)(16 30)(17 22)(18 23)(19 24)(20 21)
(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 15 11 22)(2 14 12 21)(3 13 9 24)(4 16 10 23)(5 20 26 32)(6 19 27 31)(7 18 28 30)(8 17 25 29)
(1 6)(2 5)(3 8)(4 7)(9 25)(10 28)(11 27)(12 26)(13 29)(14 32)(15 31)(16 30)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (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), (1,15,11,22)(2,14,12,21)(3,13,9,24)(4,16,10,23)(5,20,26,32)(6,19,27,31)(7,18,28,30)(8,17,25,29), (1,6)(2,5)(3,8)(4,7)(9,25)(10,28)(11,27)(12,26)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (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), (1,15,11,22)(2,14,12,21)(3,13,9,24)(4,16,10,23)(5,20,26,32)(6,19,27,31)(7,18,28,30)(8,17,25,29), (1,6)(2,5)(3,8)(4,7)(9,25)(10,28)(11,27)(12,26)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,12),(6,9),(7,10),(8,11),(13,31),(14,32),(15,29),(16,30),(17,22),(18,23),(19,24),(20,21)], [(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)], [(1,15,11,22),(2,14,12,21),(3,13,9,24),(4,16,10,23),(5,20,26,32),(6,19,27,31),(7,18,28,30),(8,17,25,29)], [(1,6),(2,5),(3,8),(4,7),(9,25),(10,28),(11,27),(12,26),(13,29),(14,32),(15,31),(16,30),(17,24),(18,23),(19,22),(20,21)]])
C2×C4⋊D4 is a maximal subgroup of
C24.(C2×C4) C24.54D4 C24.56D4 C24.60D4 C23.38D8 C24.74D4 C23.23D8 C24.76D4 M4(2)⋊20D4 C24.175C23 C23⋊2D8 C23⋊3SD16 C24.83D4 C24.84D4 C42⋊13D4 C24.198C23 C23.215C24 C24.215C23 C24.217C23 C24.218C23 C23.259C24 C24.244C23 C23.308C24 C24⋊8D4 C24.249C23 C23.316C24 C24.254C23 C23.322C24 C23.324C24 C24.258C23 C24.259C23 C23.327C24 C23.328C24 C24.262C23 C24.263C23 C23.333C24 C24.565C23 C24.269C23 C23.344C24 C23.356C24 C23.364C24 C24.293C23 C23.390C24 C23.391C24 C23.400C24 C23.401C24 C23.404C24 C42⋊17D4 C42⋊18D4 C23.439C24 C42⋊19D4 C42⋊20D4 C23.443C24 C23.455C24 C24.331C23 C23.491C24 C24.360C23 C24⋊10D4 C24.587C23 C42⋊27D4 C42⋊28D4 C23.524C24 C23.535C24 C24.592C23 C23.556C24 C24.377C23 C23.568C24 C23.569C24 C23.571C24 C23.572C24 C23.573C24 C24.384C23 C23.576C24 C23.578C24 C23.581C24 C24.389C23 C24.393C23 C24.395C23 C23.591C24 C24.406C23 C24.407C23 C23.603C24 C23.608C24 C23.611C24 C24.413C23 C24.459C23 C23.715C24 C23.716C24 C24⋊13D4 C42⋊46D4 C24.598C23 C42⋊47D4 C24.105D4 M4(2)⋊14D4 C24.117D4 C23⋊3D8 C23⋊4SD16 C24.121D4 C24.125D4 C24.126D4 C24.127D4 C2×D42 C22.77C25 C22.83C25 C4⋊2+ 1+4 C22.94C25 C22.108C25 C22.123C25 C22.125C25 C22.126C25 C22.131C25
C2×C4⋊D4 is a maximal quotient of
C42⋊15D4 C23.308C24 C24⋊8D4 C23.311C24 C24.95D4 C23.313C24 C24.249C23 C23.315C24 C23.316C24 C24.252C23 C23.324C24 C24.258C23 C24.259C23 C23.327C24 C23.328C24 C23.329C24 C24.299C23 C23.434C24 C42⋊17D4 C42.165D4 C42⋊18D4 C42.166D4 C42⋊20D4 C23.443C24 C42⋊21D4 C42.168D4 C42.169D4 C42.170D4 C42.171D4 C24⋊10D4 C24.587C23 C42⋊27D4 C42⋊28D4 C42.186D4 C24⋊13D4 C42⋊46D4 C24.598C23 C42⋊47D4 C42.440D4 C42.443D4 C42.211D4 C42.212D4 C42.444D4 C42.445D4 C42.446D4 C42.14C23 C42.15C23 C42.16C23 C42.17C23 C42.18C23 C42.19C23 C24.144D4 C24.110D4 M4(2)⋊14D4 M4(2)⋊15D4 (C2×C8)⋊11D4 (C2×C8)⋊12D4 C8.D4⋊C2 (C2×C8)⋊13D4 (C2×C8)⋊14D4 M4(2)⋊16D4 M4(2)⋊17D4 M4(2).10C23 M4(2).37D4 M4(2).38D4
Matrix representation of C2×C4⋊D4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,3,0,0,0,0,1,2,0,0,0,0,0,1,1,0,0,0,3,4],[1,0,0,0,0,0,1,4,0,0,0,2,4,0,0,0,0,0,4,4,0,0,0,0,1],[1,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,1,0,0,0,0,4] >;
C2×C4⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes D_4
% in TeX
G:=Group("C2xC4:D4"); // GroupNames label
G:=SmallGroup(64,203);
// by ID
G=gap.SmallGroup(64,203);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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