p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22.4D4, C22.13C23, C23.10C22, C4⋊C4⋊4C2, C2.7(C2×D4), C22⋊C4⋊4C2, (C22×C4)⋊3C2, (C2×D4).4C2, C2.6(C4○D4), (C2×C4).13C22, SmallGroup(32,30)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.D4
G = < a,b,c,d | a2=b2=c4=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=bc-1 >
Character table of C22.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 16 points - transitive group 16T37
(1 16)(2 7)(3 14)(4 5)(6 10)(8 12)(9 15)(11 13)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 9)(4 11)(5 7)(6 16)(8 14)(13 15)
G:=sub<Sym(16)| (1,16)(2,7)(3,14)(4,5)(6,10)(8,12)(9,15)(11,13), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,9)(4,11)(5,7)(6,16)(8,14)(13,15)>;
G:=Group( (1,16)(2,7)(3,14)(4,5)(6,10)(8,12)(9,15)(11,13), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,9)(4,11)(5,7)(6,16)(8,14)(13,15) );
G=PermutationGroup([[(1,16),(2,7),(3,14),(4,5),(6,10),(8,12),(9,15),(11,13)], [(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,9),(4,11),(5,7),(6,16),(8,14),(13,15)]])
G:=TransitiveGroup(16,37);
►On 16 points - transitive group 16T54
(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 10)(2 6)(3 12)(4 8)(5 15)(7 13)(9 14)(11 16)
G:=sub<Sym(16)| (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), (1,10)(2,6)(3,12)(4,8)(5,15)(7,13)(9,14)(11,16)>;
G:=Group( (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), (1,10)(2,6)(3,12)(4,8)(5,15)(7,13)(9,14)(11,16) );
G=PermutationGroup([[(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)], [(1,10),(2,6),(3,12),(4,8),(5,15),(7,13),(9,14),(11,16)]])
G:=TransitiveGroup(16,54);
C22.D4 is a maximal subgroup of
C23.36C23 C23⋊3D4 C22.32C24 C22.33C24 C22.34C24 C22.36C24 C22.45C24 C22.46C24 C22.47C24 C22.53C24 C22.54C24 C22.56C24 C22.57C24 C62.9D4
C23.D2p: C23.D4 C23.7D4 C23.9D6 C23.21D6 C23.28D6 C23.23D6 D10.12D4 C22.D20 ...
C2p.(C2×D4): C22.19C24 C23.38C23 D4⋊5D4 D4⋊6D4 D6.D4 D10.13D4 D14.5D4 D22.5D4 ...
C22.D4 is a maximal quotient of
C62.9D4
C23.D2p: C23.34D4 C23.8Q8 C23.23D4 C23.10D4 C23.11D4 C22.D8 C23.46D4 C23.19D4 ...
(C2×C4).D2p: C23.63C23 C24.C22 C23.81C23 C23.4Q8 C23.83C23 D6.D4 D10.13D4 D14.5D4 ...
Matrix representation of C22.D4 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 2 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,4,0,0,2,4,0,0,0,0,0,4,0,0,4,0],[1,4,0,0,0,4,0,0,0,0,1,0,0,0,0,4] >;
C22.D4 in GAP, Magma, Sage, TeX
C_2^2.D_4
% in TeX
G:=Group("C2^2.D4"); // GroupNames label
G:=SmallGroup(32,30);
// by ID
G=gap.SmallGroup(32,30);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,101,302,42]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=b*c^-1>;
// generators/relations
Export
Subgroup lattice of C22.D4 in TeX
Character table of C22.D4 in TeX