p-group, non-abelian, nilpotent (class 5), monomial
Aliases: C42.1D4, 2+ 1+4⋊C4, C2.8C2≀C4, C4.D4⋊C4, (C2×D4).1D4, C4.D8⋊1C2, C42⋊C4⋊1C2, D4⋊4D4.1C2, C4⋊1D4.1C22, C22.1(C23⋊C4), (C2×D4).1(C2×C4), (C2×C4).5(C22⋊C4), SmallGroup(128,134)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.D4
G = < a,b,c,d | a4=b4=c4=1, d2=a-1, ab=ba, cac-1=a-1b, ad=da, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a-1c-1 >
Subgroups: 232 in 61 conjugacy classes, 14 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C2×C8, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C23⋊C4, C4.D4, C4≀C2, C4⋊C8, C4⋊1D4, C8⋊C22, 2+ 1+4, C4.D8, C42⋊C4, D4⋊4D4, C42.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C23⋊C4, C2≀C4, C42.D4
Character table of C42.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 4 | 4 | 4 | 8 | 16 | 16 | 8 | 8 | 8 | 8 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | i | i | -i | -i | i | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | -i | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | -i | -i | i | i | -i | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | i | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ12 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ13 | 4 | 4 | -4 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C4 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | orthogonal faithful |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | orthogonal faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | complex faithful |
►On 16 points - transitive group 16T335
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 15 13 11)(10 16 14 12)
(1 7 3 5)(2 6 4 8)(9 11 13 15)(10 12 14 16)
(1 16)(2 13 6 15)(3 12)(4 9 8 11)(5 14 7 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,7,3,5)(2,6,4,8)(9,11,13,15)(10,12,14,16), (1,16)(2,13,6,15)(3,12)(4,9,8,11)(5,14,7,10), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,15,13,11)(10,16,14,12), (1,7,3,5)(2,6,4,8)(9,11,13,15)(10,12,14,16), (1,16)(2,13,6,15)(3,12)(4,9,8,11)(5,14,7,10), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,15,13,11),(10,16,14,12)], [(1,7,3,5),(2,6,4,8),(9,11,13,15),(10,12,14,16)], [(1,16),(2,13,6,15),(3,12),(4,9,8,11),(5,14,7,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,335);
►On 16 points - transitive group 16T343
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 8 5 4)(2 3 6 7)(9 15 13 11)(10 16 14 12)
(1 14 8 12)(2 13)(3 11 7 15)(4 16 5 10)(6 9)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (9,15,13,11)(10,16,14,12), (1,8,5,4)(2,3,6,7)(9,15,13,11)(10,16,14,12), (1,14,8,12)(2,13)(3,11,7,15)(4,16,5,10)(6,9), (1,2)(3,4)(5,6)(7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (9,15,13,11)(10,16,14,12), (1,8,5,4)(2,3,6,7)(9,15,13,11)(10,16,14,12), (1,14,8,12)(2,13)(3,11,7,15)(4,16,5,10)(6,9), (1,2)(3,4)(5,6)(7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(9,15,13,11),(10,16,14,12)], [(1,8,5,4),(2,3,6,7),(9,15,13,11),(10,16,14,12)], [(1,14,8,12),(2,13),(3,11,7,15),(4,16,5,10),(6,9)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,343);
►On 16 points - transitive group 16T387
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 4 2 3)(5 8 7 6)(9 15 13 11)(10 16 14 12)
(1 14 7 13)(2 10 5 9)(3 16 6 11)(4 12 8 15)
(3 4)(5 6)(7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,15,13,11)(10,16,14,12), (1,14,7,13)(2,10,5,9)(3,16,6,11)(4,12,8,15), (3,4)(5,6)(7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (9,15,13,11)(10,16,14,12), (1,4,2,3)(5,8,7,6)(9,15,13,11)(10,16,14,12), (1,14,7,13)(2,10,5,9)(3,16,6,11)(4,12,8,15), (3,4)(5,6)(7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(9,15,13,11),(10,16,14,12)], [(1,4,2,3),(5,8,7,6),(9,15,13,11),(10,16,14,12)], [(1,14,7,13),(2,10,5,9),(3,16,6,11),(4,12,8,15)], [(3,4),(5,6),(7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,387);
Matrix representation of C42.D4 ►in GL4(𝔽3) generated by
| 1 | 2 | 1 | 0 |
| 2 | 0 | 0 | 1 |
| 2 | 2 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 |
| 2 | 1 | 2 | 0 |
| 1 | 0 | 0 | 2 |
| 0 | 0 | 1 | 2 |
| 1 | 1 | 1 | 2 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(3))| [1,2,2,2,2,0,2,1,1,0,0,0,0,1,0,0],[1,0,2,1,0,1,1,0,0,1,2,0,1,1,0,2],[0,1,0,0,0,1,2,0,1,1,0,2,2,2,0,2],[0,1,0,0,0,0,1,0,0,0,0,1,1,1,0,1] >;
C42.D4 in GAP, Magma, Sage, TeX
C_4^2.D_4
% in TeX
G:=Group("C4^2.D4"); // GroupNames label
G:=SmallGroup(128,134);
// by ID
G=gap.SmallGroup(128,134);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1242,745,1684,1411,718,375,172,4037,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^-1,a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*c^-1>;
// generators/relations
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