p-group, metabelian, nilpotent (class 5), monomial
Aliases: (C4×C8)⋊4C4, C4⋊1D4⋊2C4, (C2×D4).5D4, C8⋊4D4.2C2, C42⋊C4⋊2C2, C42.15(C2×C4), C4⋊1D4.2C22, C2.8(C42⋊C4), C22.18(C23⋊C4), (C2×C4).34(C22⋊C4), SmallGroup(128,140)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊1D4⋊C4
G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a2b-1, dcd-1=ac >
Subgroups: 256 in 60 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C42, C22⋊C4, C2×C8, D8, C2×D4, C2×D4, C4×C8, C23⋊C4, C4⋊1D4, C2×D8, C42⋊C4, C8⋊4D4, C4⋊1D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C23⋊C4, C42⋊C4, C4⋊1D4⋊C4
Character table of C4⋊1D4⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 8 | 8 | 16 | 4 | 4 | 4 | 16 | 16 | 16 | 16 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | orthogonal lifted from C42⋊C4 |
| ρ12 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | orthogonal lifted from C42⋊C4 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | orthogonal faithful |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | orthogonal faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | orthogonal faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | orthogonal faithful |
►On 16 points - transitive group 16T377
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 7 6 8)(9 12 11 10)(13 16 15 14)
(1 7)(2 8)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)
(1 9 3 12)(2 11 4 10)(5 15)(6 13)(7 16 8 14)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,7,6,8)(9,12,11,10)(13,16,15,14), (1,7)(2,8)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15), (1,9,3,12)(2,11,4,10)(5,15)(6,13)(7,16,8,14)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,7,6,8)(9,12,11,10)(13,16,15,14), (1,7)(2,8)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15), (1,9,3,12)(2,11,4,10)(5,15)(6,13)(7,16,8,14) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,7,6,8),(9,12,11,10),(13,16,15,14)], [(1,7),(2,8),(3,6),(4,5),(9,14),(10,13),(11,16),(12,15)], [(1,9,3,12),(2,11,4,10),(5,15),(6,13),(7,16,8,14)]])
G:=TransitiveGroup(16,377);
►On 16 points - transitive group 16T413
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 2 4)(5 7 6 8)(9 12 11 10)(13 16 15 14)
(3 4)(5 8)(6 7)(9 11)(13 14)(15 16)
(1 9 6 15)(2 11 5 13)(3 10 7 14)(4 12 8 16)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,16,15,14), (3,4)(5,8)(6,7)(9,11)(13,14)(15,16), (1,9,6,15)(2,11,5,13)(3,10,7,14)(4,12,8,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,16,15,14), (3,4)(5,8)(6,7)(9,11)(13,14)(15,16), (1,9,6,15)(2,11,5,13)(3,10,7,14)(4,12,8,16) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,2,4),(5,7,6,8),(9,12,11,10),(13,16,15,14)], [(3,4),(5,8),(6,7),(9,11),(13,14),(15,16)], [(1,9,6,15),(2,11,5,13),(3,10,7,14),(4,12,8,16)]])
G:=TransitiveGroup(16,413);
Matrix representation of C4⋊1D4⋊C4 ►in GL4(𝔽7) generated by
| 6 | 3 | 6 | 1 |
| 0 | 2 | 5 | 4 |
| 0 | 5 | 6 | 6 |
| 0 | 2 | 3 | 5 |
| 0 | 1 | 2 | 6 |
| 4 | 0 | 2 | 1 |
| 4 | 4 | 1 | 6 |
| 6 | 1 | 4 | 6 |
| 2 | 2 | 4 | 5 |
| 6 | 5 | 3 | 5 |
| 6 | 1 | 4 | 0 |
| 2 | 2 | 1 | 3 |
| 4 | 1 | 2 | 1 |
| 4 | 0 | 6 | 0 |
| 2 | 2 | 2 | 3 |
| 5 | 2 | 1 | 1 |
G:=sub<GL(4,GF(7))| [6,0,0,0,3,2,5,2,6,5,6,3,1,4,6,5],[0,4,4,6,1,0,4,1,2,2,1,4,6,1,6,6],[2,6,6,2,2,5,1,2,4,3,4,1,5,5,0,3],[4,4,2,5,1,0,2,2,2,6,2,1,1,0,3,1] >;
C4⋊1D4⋊C4 in GAP, Magma, Sage, TeX
C_4\rtimes_1D_4\rtimes C_4
% in TeX
G:=Group("C4:1D4:C4"); // GroupNames label
G:=SmallGroup(128,140);
// by ID
G=gap.SmallGroup(128,140);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1059,520,794,745,1684,1411,375,172,4037]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^2=d^4=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*c>;
// generators/relations
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