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G = C2×C23⋊A4  order 192 = 26·3

Direct product of C2 and C23⋊A4

direct product, non-abelian, soluble, monomial

Aliases: C2×C23⋊A4, C24⋊6A4, 2+ (1+4)⋊5C6, Q8⋊2(C2×A4), (C2×Q8)⋊4A4, C23⋊2(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

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

Generators and relations
 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 >

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

(1 15)(2 10)(3 6)(4 11)(5 7)(8 9)(12 13)(14 16)

(1 16)(2 8)(3 7)(4 12)(5 6)(9 10)(11 13)(14 15)

(1 4)(2 3)(5 9)(6 10)(7 8)(11 15)(12 16)(13 14)

(5 9)(7 8)(12 16)(13 14)

(5 9)(6 10)(11 15)(13 14)

(5 6 7)(8 9 10)(11 12 13)(14 15 16)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,13)(6,11)(7,12)(8,16)(9,14)(10,15), (1,15)(2,10)(3,6)(4,11)(5,7)(8,9)(12,13)(14,16), (1,16)(2,8)(3,7)(4,12)(5,6)(9,10)(11,13)(14,15), (1,4)(2,3)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14), (5,9)(7,8)(12,16)(13,14), (5,9)(6,10)(11,15)(13,14), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;
 
G:=Group( (1,2)(3,4)(5,13)(6,11)(7,12)(8,16)(9,14)(10,15), (1,15)(2,10)(3,6)(4,11)(5,7)(8,9)(12,13)(14,16), (1,16)(2,8)(3,7)(4,12)(5,6)(9,10)(11,13)(14,15), (1,4)(2,3)(5,9)(6,10)(7,8)(11,15)(12,16)(13,14), (5,9)(7,8)(12,16)(13,14), (5,9)(6,10)(11,15)(13,14), (5,6,7)(8,9,10)(11,12,13)(14,15,16) );
 
G=PermutationGroup([(1,2),(3,4),(5,13),(6,11),(7,12),(8,16),(9,14),(10,15)], [(1,15),(2,10),(3,6),(4,11),(5,7),(8,9),(12,13),(14,16)], [(1,16),(2,8),(3,7),(4,12),(5,6),(9,10),(11,13),(14,15)], [(1,4),(2,3),(5,9),(6,10),(7,8),(11,15),(12,16),(13,14)], [(5,9),(7,8),(12,16),(13,14)], [(5,9),(6,10),(11,15),(13,14)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)])
 
G:=TransitiveGroup(16,424);
 

Matrix representation ►G ⊆ GL7(ℤ)

-1000000
0-100000
00-10000
000-1000
0000-100
00000-10
000000-1
,
0100000
1000000
-1-1-10000
0000001
0000010
0000100
0001000
,
-1-1-10000
0010000
0100000
0000100
0001000
0000001
0000010
,
1000000
0100000
0010000
000-1000
0000-100
00000-10
000000-1
,
0010000
-1-1-10000
1000000
000000-1
0000010
0000100
000-1000
,
0100000
1000000
-1-1-10000
0000-100
000-1000
0000001
0000010
,
1000000
0010000
-1-1-10000
0001000
0000001
0000100
0000010

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] >;
 

Character table of C2×C23⋊A4

 class 12A2B2C2D2E2F2G2H2I3A3B4A4B4C4D6A6B6C6D6E6F
 size 111166666616166666161616161616
ρ11111111111111111111111    trivial
ρ21-11-1111-1-1-111-111-111-1-1-1-1    linear of order 2
ρ31-11-1111-1-1-1ζ3ζ32-111-1ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ41111111111ζ3ζ321111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ51111111111ζ32ζ31111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ61-11-1111-1-1-1ζ32ζ3-111-1ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ73-33-3-1-131-31001-1-11000000    orthogonal lifted from C2×A4
ρ83333-1-1-1-1-1-100-13-13000000    orthogonal lifted from A4
ρ93-33-3-13-1-311001-1-11000000    orthogonal lifted from C2×A4
ρ103-33-33-1-111-3001-1-11000000    orthogonal lifted from C2×A4
ρ113333-1-1-1-1-1-1003-13-1000000    orthogonal lifted from A4
ρ123-33-3-1-1-111100-3-131000000    orthogonal lifted from C2×A4
ρ1333333-1-1-1-1300-1-1-1-1000000    orthogonal lifted from A4
ρ143333-13-13-1-100-1-1-1-1000000    orthogonal lifted from A4
ρ153-33-3-1-1-11110013-1-3000000    orthogonal lifted from C2×A4
ρ163333-1-13-13-100-1-1-1-1000000    orthogonal lifted from A4
ρ1744-4-4000000110000-1-11-11-1    orthogonal lifted from C23⋊A4
ρ184-4-44000000110000-1-1-11-11    orthogonal lifted from C23⋊A4
ρ194-4-44000000ζ32ζ30000ζ6ζ65ζ65ζ3ζ6ζ32    complex lifted from C23⋊A4
ρ2044-4-4000000ζ3ζ320000ζ65ζ6ζ32ζ6ζ3ζ65    complex lifted from C23⋊A4
ρ214-4-44000000ζ3ζ320000ζ65ζ6ζ6ζ32ζ65ζ3    complex lifted from C23⋊A4
ρ2244-4-4000000ζ32ζ30000ζ6ζ65ζ3ζ65ζ32ζ6    complex lifted from 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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