direct product, p-group, abelian, monomial
Aliases: C22×C4, SmallGroup(16,10)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C22×C4 |
| C1 — C22×C4 |
| C1 — C22×C4 |
Generators and relations for C22×C4
G = < a,b,c | a2=b2=c4=1, ab=ba, ac=ca, bc=cb >
Character table of C22×C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 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 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 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 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ10 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | -i | i | -i | i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ14 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | i | -i | i | -i | -i | i | linear of order 4 |
►Regular action on 16 points - transitive group 16T2
Generators in S16
(1 8)(2 5)(3 6)(4 7)(9 15)(10 16)(11 13)(12 14)
(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)| (1,8)(2,5)(3,6)(4,7)(9,15)(10,16)(11,13)(12,14), (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( (1,8)(2,5)(3,6)(4,7)(9,15)(10,16)(11,13)(12,14), (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([[(1,8),(2,5),(3,6),(4,7),(9,15),(10,16),(11,13),(12,14)], [(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(16,2);
C22×C4 is a maximal subgroup of
C2.C42 C22⋊C8 C42⋊C2 C4⋊D4 C22⋊Q8 C22.D4
C22×C4 is a maximal quotient of C42⋊C2 C8○D4
| action | f(x) | Disc(f) |
|---|---|---|
| 16T2 | x16-x12+x8-x4+1 | 232·512 |
Matrix representation of C22×C4 ►in GL3(𝔽5) generated by
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
G:=sub<GL(3,GF(5))| [1,0,0,0,4,0,0,0,4],[4,0,0,0,1,0,0,0,1],[1,0,0,0,4,0,0,0,2] >;
C22×C4 in GAP, Magma, Sage, TeX
C_2^2\times C_4
% in TeX
G:=Group("C2^2xC4"); // GroupNames label
G:=SmallGroup(16,10);
// by ID
G=gap.SmallGroup(16,10);
# by ID
G:=PCGroup([4,-2,2,2,-2,32]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^4=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of C22×C4 in TeX
Character table of C22×C4 in TeX