p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C23⋊3D4, C24⋊2C22, C23.10C23, C22.28C24, C2.32+ 1+4, C4⋊D4⋊5C2, C4⋊C4⋊3C22, C22≀C2⋊2C2, (C22×D4)⋊6C2, (C2×D4)⋊13C22, C22⋊C4⋊3C22, (C2×C4).16C23, (C22×C4)⋊7C22, C22.20(C2×D4), C2.13(C22×D4), C22.D4⋊2C2, (C2×C22⋊C4)⋊12C2, SmallGroup(64,215)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23⋊3D4
G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 321 in 173 conjugacy classes, 81 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, C23⋊3D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4
Character table of C23⋊3D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 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 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►On 16 points - transitive group 16T87
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 3)(2 16)(4 14)(5 7)(6 11)(8 9)(10 12)(13 15)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 2)(3 4)(5 9)(6 12)(7 11)(8 10)(13 14)(15 16)
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,3)(2,16)(4,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,9)(6,12)(7,11)(8,10)(13,14)(15,16)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,3)(2,16)(4,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,9)(6,12)(7,11)(8,10)(13,14)(15,16) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(1,3),(2,16),(4,14),(5,7),(6,11),(8,9),(10,12),(13,15)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,2),(3,4),(5,9),(6,12),(7,11),(8,10),(13,14),(15,16)]])
G:=TransitiveGroup(16,87);
►On 16 points - transitive group 16T119
(5 12)(6 9)(7 10)(8 11)
(2 14)(4 16)(5 12)(7 10)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 7)(2 6)(3 5)(4 8)(9 14)(10 13)(11 16)(12 15)
G:=sub<Sym(16)| (5,12)(6,9)(7,10)(8,11), (2,14)(4,16)(5,12)(7,10), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15)>;
G:=Group( (5,12)(6,9)(7,10)(8,11), (2,14)(4,16)(5,12)(7,10), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15) );
G=PermutationGroup([[(5,12),(6,9),(7,10),(8,11)], [(2,14),(4,16),(5,12),(7,10)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,7),(2,6),(3,5),(4,8),(9,14),(10,13),(11,16),(12,15)]])
G:=TransitiveGroup(16,119);
C23⋊3D4 is a maximal subgroup of
C23⋊2SD16 C23.5D8 C24.16D4 C24.28D4 C24.31D4 C24.33D4 C24.36D4 C22.38C25 C22.73C25 C22.74C25 C22.79C25 C22.80C25 C4⋊2+ 1+4 C42⋊C23 C22.122C25 C22.123C25 C22.134C25 C22.149C25
C23⋊D4p: C23⋊D8 C23⋊4D12 C23⋊3D20 C23⋊3D28 ...
C24⋊D2p: C24⋊D4 C24⋊C23 C24⋊7D6 C24⋊12D6 C24⋊3D10 C24⋊8D10 C24⋊2D14 C24⋊7D14 ...
C2p.2+ 1+4: C22.48C25 C22.94C25 C22.126C25 C22.131C25 C22.132C25 C22.147C25 C6.372+ 1+4 C6.1202+ 1+4 ...
C23⋊3D4 is a maximal quotient of
C24.90D4 C24.91D4 C23.203C24 C24.195C23 C24.198C23 C24.94D4 C24.95D4 C24.96D4 C23.434C24 C23.439C24 C23.443C24 C23.461C24 C24.583C23 C23.479C24 C23.483C24 C23.491C24 C23.500C24 C23.502C24 C24.587C23 C24.97D4 C24⋊5Q8 C23.527C24 C23.530C24 C23.535C24 C24.374C23 C24.592C23 C23.556C24 C23.559C24 C24.377C23 C24.378C23 C23.568C24 C23.569C24 C23.571C24 C23.573C24 C23.578C24 C25⋊C22 C24.389C23 C23.584C24 C24.393C23 C24.395C23 C23.597C24 C24.406C23 C24.407C23 C23.608C24 C24.411C23 C23.617C24 C23.624C24 C24.420C23 C24.421C23 C23.630C24 C23.631C24 C23.632C24 C23.633C24 C23.634C24 C24.459C23 C23.714C24 C23.715C24 C23.716C24 C24.462C23
C24⋊D2p: C24⋊7D4 C24⋊8D4 C24⋊9D4 C24⋊10D4 C24⋊11D4 C24⋊7D6 C24⋊12D6 C24⋊3D10 ...
C2p.2+ 1+4: C23⋊3D8 C23⋊4SD16 C24.121D4 C23⋊3Q16 C24.123D4 C24.124D4 C24.125D4 C24.126D4 ...
Matrix representation of C23⋊3D4 ►in GL6(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 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 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C23⋊3D4 in GAP, Magma, Sage, TeX
C_2^3\rtimes_3D_4
% in TeX
G:=Group("C2^3:3D4"); // GroupNames label
G:=SmallGroup(64,215);
// by ID
G=gap.SmallGroup(64,215);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,650,188,579]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=e^2=1,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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