p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C4, C2.1D4, C23.C2, C22.2C22, (C2×C4)⋊1C2, C2.1(C2×C4), SmallGroup(16,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22⋊C4
G = < a,b,c | a2=b2=c4=1, cac-1=ab=ba, bc=cb >
Character table of C22⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
►On 8 points - transitive group 8T10
(2 6)(4 8)
(1 5)(2 6)(3 7)(4 8)
(1 2 3 4)(5 6 7 8)
G:=sub<Sym(8)| (2,6)(4,8), (1,5)(2,6)(3,7)(4,8), (1,2,3,4)(5,6,7,8)>;
G:=Group( (2,6)(4,8), (1,5)(2,6)(3,7)(4,8), (1,2,3,4)(5,6,7,8) );
G=PermutationGroup([[(2,6),(4,8)], [(1,5),(2,6),(3,7),(4,8)], [(1,2,3,4),(5,6,7,8)]])
G:=TransitiveGroup(8,10);
►Regular action on 16 points - transitive group 16T10
(1 6)(2 12)(3 8)(4 10)(5 14)(7 16)(9 13)(11 15)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,6)(2,12)(3,8)(4,10)(5,14)(7,16)(9,13)(11,15), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,6)(2,12)(3,8)(4,10)(5,14)(7,16)(9,13)(11,15), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,6),(2,12),(3,8),(4,10),(5,14),(7,16),(9,13),(11,15)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,10);
C22⋊C4 is a maximal subgroup of
C23⋊C4 C42⋊C2 C4×D4 C22≀C2 C4⋊D4 C22.D4 C42⋊2C2 A4⋊C4 S32⋊C4 C62⋊C4 D52⋊C4
C2p.D4: C22⋊Q8 C4.4D4 D6⋊C4 C6.D4 D10⋊C4 C23.D5 D14⋊C4 C23.D7 ...
Dp.D4, p=1 mod 4: C22⋊F5 D13.D4 D17.D4 D29.D4 ...
C22⋊C4 is a maximal quotient of
C2.C42 C23⋊C4 S32⋊C4 C62⋊C4 D52⋊C4
D2p⋊C4: D4⋊C4 C4≀C2 D6⋊C4 D10⋊C4 C22⋊F5 D14⋊C4 D22⋊C4 D26⋊C4 ...
C2p.D4: C22⋊C8 C4.D4 C4.10D4 Q8⋊C4 C6.D4 C23.D5 C23.D7 C23.D11 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T10 | x8-13x6+44x4-17x2+1 | 216·34·56·292 |
Matrix representation of C22⋊C4 ►in GL3(𝔽5) generated by
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 3 | 1 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 2 | 0 | 0 |
| 0 | 2 | 3 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(5))| [1,0,0,0,4,3,0,0,1],[1,0,0,0,4,0,0,0,4],[2,0,0,0,2,0,0,3,3] >;
C22⋊C4 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4
% in TeX
G:=Group("C2^2:C4"); // GroupNames label
G:=SmallGroup(16,3);
// by ID
G=gap.SmallGroup(16,3);
# by ID
G:=PCGroup([4,-2,2,-2,2,32,49]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^4=1,c*a*c^-1=a*b=b*a,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of C22⋊C4 in TeX
Character table of C22⋊C4 in TeX