p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.2D4, (C2×C4).2D4, C22⋊C4⋊2C4, (C22×C4)⋊2C4, C4.D4.C2, C23⋊C4.1C2, C23.2(C2×C4), C2.7(C23⋊C4), (C2×D4).2C22, C22.D4.1C2, C22.10(C22⋊C4), SmallGroup(64,33)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad3 >
Character table of C23.D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | i | 1 | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | i | -i | -1 | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | i | -i | 1 | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -i | i | -1 | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T140
Generators in S16
(1 3)(2 8)(4 6)(5 7)(9 11)(10 12)(13 15)(14 16)
(2 6)(4 8)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 3 13)(2 12 8 10)(4 14 6 16)(5 11 7 9)
G:=sub<Sym(16)| (1,3)(2,8)(4,6)(5,7)(9,11)(10,12)(13,15)(14,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,3,13)(2,12,8,10)(4,14,6,16)(5,11,7,9)>;
G:=Group( (1,3)(2,8)(4,6)(5,7)(9,11)(10,12)(13,15)(14,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,3,13)(2,12,8,10)(4,14,6,16)(5,11,7,9) );
G=PermutationGroup([[(1,3),(2,8),(4,6),(5,7),(9,11),(10,12),(13,15),(14,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15,3,13),(2,12,8,10),(4,14,6,16),(5,11,7,9)]])
G:=TransitiveGroup(16,140);
►On 16 points - transitive group 16T148
Generators in S16
(2 10)(3 7)(4 16)(6 14)(8 12)(11 15)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 16 10 4)(3 15 7 11)(6 12 14 8)(9 13)
G:=sub<Sym(16)| (2,10)(3,7)(4,16)(6,14)(8,12)(11,15), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,16,10,4)(3,15,7,11)(6,12,14,8)(9,13)>;
G:=Group( (2,10)(3,7)(4,16)(6,14)(8,12)(11,15), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,16,10,4)(3,15,7,11)(6,12,14,8)(9,13) );
G=PermutationGroup([[(2,10),(3,7),(4,16),(6,14),(8,12),(11,15)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,16,10,4),(3,15,7,11),(6,12,14,8),(9,13)]])
G:=TransitiveGroup(16,148);
►On 16 points - transitive group 16T153
Generators in S16
(1 9)(2 10)(3 15)(4 16)(5 13)(6 14)(7 11)(8 12)
(1 5)(3 7)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 4 9 16)(2 11 10 7)(3 6 15 14)(5 8 13 12)
G:=sub<Sym(16)| (1,9)(2,10)(3,15)(4,16)(5,13)(6,14)(7,11)(8,12), (1,5)(3,7)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,9,16)(2,11,10,7)(3,6,15,14)(5,8,13,12)>;
G:=Group( (1,9)(2,10)(3,15)(4,16)(5,13)(6,14)(7,11)(8,12), (1,5)(3,7)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,9,16)(2,11,10,7)(3,6,15,14)(5,8,13,12) );
G=PermutationGroup([[(1,9),(2,10),(3,15),(4,16),(5,13),(6,14),(7,11),(8,12)], [(1,5),(3,7),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,4,9,16),(2,11,10,7),(3,6,15,14),(5,8,13,12)]])
G:=TransitiveGroup(16,153);
►On 16 points - transitive group 16T160
Generators in S16
(2 6)(3 7)(9 13)(12 16)
(2 6)(4 8)(10 14)(12 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 13 6 9)(3 12 7 16)(4 15)(5 10)(8 11)
G:=sub<Sym(16)| (2,6)(3,7)(9,13)(12,16), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13,6,9)(3,12,7,16)(4,15)(5,10)(8,11)>;
G:=Group( (2,6)(3,7)(9,13)(12,16), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13,6,9)(3,12,7,16)(4,15)(5,10)(8,11) );
G=PermutationGroup([[(2,6),(3,7),(9,13),(12,16)], [(2,6),(4,8),(10,14),(12,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,13,6,9),(3,12,7,16),(4,15),(5,10),(8,11)]])
G:=TransitiveGroup(16,160);
C23.D4 is a maximal subgroup of
C42⋊4D4 C42⋊6D4 C42.14D4 C22⋊C4⋊F5 (C22×C4)⋊F5
(C2×D4).D2p: C4○C2≀C4 C24.36D4 C23.(C2×D4) C42.13D4 (C2×D4).D6 C23.4D12 (C22×C12)⋊C4 (C2×C20).D4 ...
C23.D4 is a maximal quotient of
C23.15M4(2) C23.2M4(2) C24.4D4 (C2×C4).Q16 C22⋊C4⋊F5 (C22×C4)⋊F5
C23.D4p: C23.4D8 C23.4D12 C23.4D20 C23.4D28 ...
(C2×C4).D4p: (C2×C4).D8 (C2×D4).D6 (C2×C20).D4 (C2×C28).D4 ...
(C2×D4).D2p: C24.5D4 (C22×C12)⋊C4 (C22×C20)⋊C4 (C22×C28)⋊C4 ...
Matrix representation of C23.D4 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,1,0,0,2,0,0,3,0,0,1,0,0,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,2,0,0,0,0,0,0,1,0,0,3,0],[0,0,3,0,4,0,0,0,0,0,0,2,0,4,0,0] >;
C23.D4 in GAP, Magma, Sage, TeX
C_2^3.D_4
% in TeX
G:=Group("C2^3.D4"); // GroupNames label
G:=SmallGroup(64,33);
// by ID
G=gap.SmallGroup(64,33);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,297,255,1444]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^3>;
// generators/relations
Export
Subgroup lattice of C23.D4 in TeX
Character table of C23.D4 in TeX