p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23⋊C4, C22.2D4, C23.1C22, (C2×C4)⋊C4, C22⋊C4⋊1C2, (C2×D4).1C2, C22.2(C2×C4), C2.3(C22⋊C4), SmallGroup(32,6)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23⋊C4
G = < a,b,c,d | a2=b2=c2=d4=1, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >
Character table of C23⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | i | -1 | -i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | i | 1 | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -1 | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -i | 1 | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T19
Generators in S8
(1 2)(3 5)(4 8)(6 7)
(1 3)(2 5)(4 7)(6 8)
(1 6)(2 7)(3 8)(4 5)
(1 2 3 4)(5 6 7 8)
G:=sub<Sym(8)| (1,2)(3,5)(4,8)(6,7), (1,3)(2,5)(4,7)(6,8), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8)>;
G:=Group( (1,2)(3,5)(4,8)(6,7), (1,3)(2,5)(4,7)(6,8), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8) );
G=PermutationGroup([[(1,2),(3,5),(4,8),(6,7)], [(1,3),(2,5),(4,7),(6,8)], [(1,6),(2,7),(3,8),(4,5)], [(1,2,3,4),(5,6,7,8)]])
G:=TransitiveGroup(8,19);
►On 8 points - transitive group 8T20
Generators in S8
(2 7)(3 8)
(2 7)(4 5)
(1 6)(2 7)(3 8)(4 5)
(1 2 3 4)(5 6 7 8)
G:=sub<Sym(8)| (2,7)(3,8), (2,7)(4,5), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8)>;
G:=Group( (2,7)(3,8), (2,7)(4,5), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8) );
G=PermutationGroup([[(2,7),(3,8)], [(2,7),(4,5)], [(1,6),(2,7),(3,8),(4,5)], [(1,2,3,4),(5,6,7,8)]])
G:=TransitiveGroup(8,20);
►On 8 points - transitive group 8T21
Generators in S8
(1 5)(2 6)(3 8)(4 7)
(1 4)(5 7)
(1 4)(2 3)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)
G:=sub<Sym(8)| (1,5)(2,6)(3,8)(4,7), (1,4)(5,7), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8)>;
G:=Group( (1,5)(2,6)(3,8)(4,7), (1,4)(5,7), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8) );
G=PermutationGroup([[(1,5),(2,6),(3,8),(4,7)], [(1,4),(5,7)], [(1,4),(2,3),(5,7),(6,8)], [(1,2),(3,4),(5,6,7,8)]])
G:=TransitiveGroup(8,21);
►On 16 points - transitive group 16T33
Generators in S16
(1 10)(2 9)(3 5)(4 8)(6 16)(7 15)(11 14)(12 13)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,10)(2,9)(3,5)(4,8)(6,16)(7,15)(11,14)(12,13), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,10)(2,9)(3,5)(4,8)(6,16)(7,15)(11,14)(12,13), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,10),(2,9),(3,5),(4,8),(6,16),(7,15),(11,14),(12,13)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,33);
►On 16 points - transitive group 16T52
Generators in S16
(2 11)(3 13)(4 6)(5 12)(8 16)(9 14)
(1 7)(2 11)(3 5)(4 9)(6 14)(8 16)(10 15)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (2,11)(3,13)(4,6)(5,12)(8,16)(9,14), (1,7)(2,11)(3,5)(4,9)(6,14)(8,16)(10,15)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (2,11)(3,13)(4,6)(5,12)(8,16)(9,14), (1,7)(2,11)(3,5)(4,9)(6,14)(8,16)(10,15)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(2,11),(3,13),(4,6),(5,12),(8,16),(9,14)], [(1,7),(2,11),(3,5),(4,9),(6,14),(8,16),(10,15),(12,13)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,52);
►On 16 points - transitive group 16T53
Generators in S16
(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,10),(2,8),(3,5),(4,9),(6,14),(7,15),(11,16),(12,13)], [(2,16),(4,14),(6,9),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,53);
C23⋊C4 is a maximal subgroup of
C23.C23 C2≀C22 D10.D4 C23⋊F5 C62.2D4 (C6×C12)⋊C4 (C2×C62)⋊C4 D26.D4 D26.4D4
C23.D2p: C2≀C4 C23.D4 C42⋊C4 C42⋊3C4 C23.7D4 C23.6D6 C23.7D6 C23.1D10 ...
C23⋊C4 is a maximal quotient of
C23⋊C8 C22.M4(2) C42.C4 C42.3C4 D10.D4 C23⋊F5 C62.2D4 (C6×C12)⋊C4 (C2×C62)⋊C4 D26.D4 D26.4D4
C23.D2p: C22.SD16 C23.31D4 C23.9D4 C2≀C4 C23.D4 C42⋊C4 C42⋊3C4 C23.6D6 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T19 | x8-14x6+36x4-28x2+4 | 232·174 |
| 8T20 | x8-4x7-5x6+24x5+14x4-36x3-20x2+6x+1 | 28·56·612 |
| 8T21 | x8-14x6+39x4-32x2+8 | 227·174 |
Matrix representation of C23⋊C4 ►in GL4(ℤ) generated by
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C23⋊C4 in GAP, Magma, Sage, TeX
C_2^3\rtimes C_4
% in TeX
G:=Group("C2^3:C4"); // GroupNames label
G:=SmallGroup(32,6);
// by ID
G=gap.SmallGroup(32,6);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,248]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export
Subgroup lattice of C23⋊C4 in TeX
Character table of C23⋊C4 in TeX