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## G = C2×C22⋊C4order 32 = 25

### Direct product of C2 and C22⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C22⋊C4, C24.C2, C232C4, C22.12D4, C22.4C23, C23.6C22, C2.1(C2×D4), C222(C2×C4), (C22×C4)⋊1C2, (C2×C4)⋊3C22, C2.1(C22×C4), SmallGroup(32,22)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C22⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4
 Lower central C1 — C2 — C2×C22⋊C4
 Upper central C1 — C23 — C2×C22⋊C4
 Jennings C1 — C22 — C2×C22⋊C4

Generators and relations for C2×C22⋊C4
G = < a,b,c,d | a2=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 94 in 66 conjugacy classes, 38 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C24, C2×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4

Character table of C2×C22⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 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 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 -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 -1 1 -1 linear of order 2 ρ5 1 1 1 1 -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 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 -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 1 -1 -1 1 linear of order 2 ρ9 1 -1 -1 1 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 -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 -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 -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 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 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 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 1 -1 -1 1 -i i -i i i -i i -i linear of order 4 ρ17 2 -2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4

Permutation representations of C2×C22⋊C4
On 16 points - transitive group 16T21
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)
(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)

G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (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)>;

G:=Group( (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15), (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) );

G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15)], [(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)]])

G:=TransitiveGroup(16,21);

Matrix representation of C2×C22⋊C4 in GL4(𝔽5) generated by

 1 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 4 0 0 0 0 1 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 2 0 0 0 0 4 0 0 0 0 0 4 0 0 4 0
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,4,0,0,0,0,0,4,0,0,4,0] >;

C2×C22⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes C_4
% in TeX

G:=Group("C2xC2^2:C4");
// GroupNames label

G:=SmallGroup(32,22);
// by ID

G=gap.SmallGroup(32,22);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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