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## G = C22×C4order 16 = 24

### Abelian group of type [2,2,4]

Aliases: C22×C4, SmallGroup(16,10)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4
 Chief series C1 — C2 — C22 — C23 — C22×C4
 Lower central C1 — C22×C4
 Upper central C1 — C22×C4
 Jennings C1 — C2 — 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

Permutation representations of C22×C4
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

Polynomial with Galois group C22×C4 over ℚ
actionf(x)Disc(f)
16T2x16-x12+x8-x4+1232·512

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

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