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G = C23.S4order 192 = 26·3

4th non-split extension by C23 of S4 acting faithfully

non-abelian, soluble, monomial

Aliases: C23.4S4, 2+ 1+4.2S3, C2.4(C22⋊S4), C23⋊A4.2C2, SmallGroup(192,1491)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4C23⋊A4 — C23.S4
C1C2C232+ 1+4C23⋊A4 — C23.S4
C23⋊A4 — C23.S4
C1C2

Generators and relations for C23.S4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=1, g2=c, ab=ba, eae=ac=ca, ad=da, faf-1=gag-1=b, dbd=ebe=bc=cb, fbf-1=abc, gbg-1=a, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg-1=de=ed, fef-1=d, eg=ge, gfg-1=f-1 >

Subgroups: 357 in 71 conjugacy classes, 8 normal (5 characteristic)
C1, C2, C2 [×3], C3, C4 [×5], C22 [×7], C6, C2×C4 [×9], D4 [×3], Q8, C23 [×3], C23, Dic3, A4 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C22×C4, C2×D4 [×3], C4○D4, SL2(𝔽3), C2×A4 [×3], C23⋊C4 [×3], C22.D4 [×3], 2+ 1+4, A4⋊C4 [×3], C23.7D4, C23⋊A4, C23.S4
Quotients: C1, C2, S3, S4 [×3], C22⋊S4, C23.S4

Character table of C23.S4

 class 12A2B2C2D34A4B4C4D4E4F6
 size 116663212121224242432
ρ11111111111111    trivial
ρ21111111-1-1-1-1-11    linear of order 2
ρ322222-1200000-1    orthogonal lifted from S3
ρ4333-1-10-1-1-1-1110    orthogonal lifted from S4
ρ5333-1-10-1111-1-10    orthogonal lifted from S4
ρ633-1-130-1-1-111-10    orthogonal lifted from S4
ρ733-1-130-111-1-110    orthogonal lifted from S4
ρ833-13-10-111-11-10    orthogonal lifted from S4
ρ933-13-10-1-1-11-110    orthogonal lifted from S4
ρ104-400010-2i2i000-1    complex faithful
ρ114-4000102i-2i000-1    complex faithful
ρ1266-2-2-202000000    orthogonal lifted from C22⋊S4
ρ138-8000-10000001    symplectic faithful, Schur index 2

Permutation representations of C23.S4
On 16 points - transitive group 16T441
Generators in S16
(1 8)(2 16)(3 6)(4 14)(5 12)(7 10)(9 13)(11 15)
(1 15)(2 5)(3 13)(4 7)(6 9)(8 11)(10 14)(12 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 8)(2 16)(3 6)(4 14)(5 10)(7 12)(9 15)(11 13)
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)
(5 16 10)(6 11 13)(7 14 12)(8 9 15)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,441);

Matrix representation of C23.S4 in GL4(𝔽5) generated by

1100
0400
1101
4010
,
1024
3323
1444
2302
,
4000
0400
0040
0004
,
2043
1141
3033
0004
,
1213
0311
0223
0004
,
0430
0024
0010
1040
,
3000
0012
0030
0210
G:=sub<GL(4,GF(5))| [1,0,1,4,1,4,1,0,0,0,0,1,0,0,1,0],[1,3,1,2,0,3,4,3,2,2,4,0,4,3,4,2],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,3,0,0,1,0,0,4,4,3,0,3,1,3,4],[1,0,0,0,2,3,2,0,1,1,2,0,3,1,3,4],[0,0,0,1,4,0,0,0,3,2,1,4,0,4,0,0],[3,0,0,0,0,0,0,2,0,1,3,1,0,2,0,0] >;

C23.S4 in GAP, Magma, Sage, TeX

C_2^3.S_4
% in TeX

G:=Group("C2^3.S4");
// GroupNames label

G:=SmallGroup(192,1491);
// by ID

G=gap.SmallGroup(192,1491);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,718,1013,516,530]);
// Polycyclic

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

Export

Character table of C23.S4 in TeX

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