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## G = C2×C23⋊A4order 192 = 26·3

### Direct product of C2 and C23⋊A4

Aliases: C2×C23⋊A4, C246A4, 2+ 1+45C6, Q82(C2×A4), (C2×Q8)⋊4A4, C232(C2×A4), (C2×2+ 1+4)⋊2C3, C22.5(C22⋊A4), C2.4(C2×C22⋊A4), SmallGroup(192,1508)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — C2×C23⋊A4
 Chief series C1 — C2 — C23 — 2+ 1+4 — C23⋊A4 — C2×C23⋊A4
 Lower central 2+ 1+4 — C2×C23⋊A4
 Upper central C1 — C22

Generators and relations for C2×C23⋊A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, gbg-1=bc=cb, fbf=bd=db, be=eb, ece=cd=dc, cf=fc, gcg-1=b, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 751 in 193 conjugacy classes, 19 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C22, C22 [×22], C6 [×3], C2×C4 [×14], D4 [×24], Q8 [×2], Q8 [×2], C23 [×3], C23 [×18], A4 [×3], C2×C6, C22×C4 [×3], C2×D4 [×30], C2×Q8 [×2], C4○D4 [×16], C24 [×3], C24, SL2(𝔽3) [×2], C2×A4 [×9], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4, 2+ 1+4 [×5], C2×SL2(𝔽3) [×2], C22×A4 [×3], C2×2+ 1+4, C23⋊A4, C2×C23⋊A4
Quotients: C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C22⋊A4, C23⋊A4 [×2], C2×C22⋊A4, C2×C23⋊A4

Character table of C2×C23⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F size 1 1 1 1 6 6 6 6 6 6 16 16 6 6 6 6 16 16 16 16 16 16 ρ1 1 1 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 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 -1 -1 -1 ζ3 ζ32 -1 1 1 -1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ4 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 -1 1 -1 1 1 1 -1 -1 -1 ζ32 ζ3 -1 1 1 -1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ7 3 -3 3 -3 -1 -1 3 1 -3 1 0 0 1 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ8 3 3 3 3 -1 -1 -1 -1 -1 -1 0 0 -1 3 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ9 3 -3 3 -3 -1 3 -1 -3 1 1 0 0 1 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ10 3 -3 3 -3 3 -1 -1 1 1 -3 0 0 1 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ11 3 3 3 3 -1 -1 -1 -1 -1 -1 0 0 3 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ12 3 -3 3 -3 -1 -1 -1 1 1 1 0 0 -3 -1 3 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ13 3 3 3 3 3 -1 -1 -1 -1 3 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 3 3 -1 3 -1 3 -1 -1 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 -3 3 -3 -1 -1 -1 1 1 1 0 0 1 3 -1 -3 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 3 3 3 -1 -1 3 -1 3 -1 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ17 4 4 -4 -4 0 0 0 0 0 0 1 1 0 0 0 0 -1 -1 1 -1 1 -1 orthogonal lifted from C23⋊A4 ρ18 4 -4 -4 4 0 0 0 0 0 0 1 1 0 0 0 0 -1 -1 -1 1 -1 1 orthogonal lifted from C23⋊A4 ρ19 4 -4 -4 4 0 0 0 0 0 0 ζ32 ζ3 0 0 0 0 ζ6 ζ65 ζ65 ζ3 ζ6 ζ32 complex lifted from C23⋊A4 ρ20 4 4 -4 -4 0 0 0 0 0 0 ζ3 ζ32 0 0 0 0 ζ65 ζ6 ζ32 ζ6 ζ3 ζ65 complex lifted from C23⋊A4 ρ21 4 -4 -4 4 0 0 0 0 0 0 ζ3 ζ32 0 0 0 0 ζ65 ζ6 ζ6 ζ32 ζ65 ζ3 complex lifted from C23⋊A4 ρ22 4 4 -4 -4 0 0 0 0 0 0 ζ32 ζ3 0 0 0 0 ζ6 ζ65 ζ3 ζ65 ζ32 ζ6 complex lifted from C23⋊A4

Permutation representations of C2×C23⋊A4
On 16 points - transitive group 16T424
Generators in S16
(1 3)(2 4)(5 15)(6 16)(7 14)(8 13)(9 11)(10 12)
(1 6)(2 8)(3 16)(4 13)(5 7)(9 10)(11 12)(14 15)
(1 7)(2 9)(3 14)(4 11)(5 6)(8 10)(12 13)(15 16)
(1 2)(3 4)(5 10)(6 8)(7 9)(11 14)(12 15)(13 16)
(5 10)(7 9)(11 14)(12 15)
(5 10)(6 8)(12 15)(13 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

G:=TransitiveGroup(16,424);

Matrix representation of C2×C23⋊A4 in GL7(ℤ)

 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0
,
 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1
,
 0 0 1 0 0 0 0 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[0,1,-1,0,0,0,0,1,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0],[-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0],[0,1,-1,0,0,0,0,1,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0] >;

C2×C23⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes A_4
% in TeX

G:=Group("C2xC2^3:A4");
// GroupNames label

G:=SmallGroup(192,1508);
// by ID

G=gap.SmallGroup(192,1508);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,135,262,851,375,1524,1027]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,g*b*g^-1=b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e=c*d=d*c,c*f=f*c,g*c*g^-1=b,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations

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