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G = C23xS4order 192 = 26·3

Direct product of C23 and S4

direct product, non-abelian, soluble, monomial, rational

Aliases: C23xS4, A4:C24, C25:2S3, C24:4D6, (C2xA4):C23, C22:(S3xC23), C23:(C22xS3), (C23xA4):3C2, (C22xA4):4C22, SmallGroup(192,1537)

Series: Derived Chief Lower central Upper central

C1C22A4 — C23xS4
C1C22A4S4C2xS4C22xS4 — C23xS4
A4 — C23xS4
C1C23

Generators and relations for C23xS4
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 2398 in 701 conjugacy classes, 99 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2xC4, D4, C23, C23, C23, A4, D6, C2xC6, C22xC4, C2xD4, C24, C24, S4, C2xA4, C22xS3, C22xC6, C23xC4, C22xD4, C25, C25, C2xS4, C22xA4, S3xC23, D4xC23, C22xS4, C23xA4, C23xS4
Quotients: C1, C2, C22, S3, C23, D6, C24, S4, C22xS3, C2xS4, S3xC23, C22xS4, C23xS4

Permutation representations of C23xS4
On 24 points - transitive group 24T400
Generators in S24
(1 9)(2 7)(3 8)(4 24)(5 22)(6 23)(10 14)(11 15)(12 13)(16 20)(17 21)(18 19)
(1 23)(2 24)(3 22)(4 7)(5 8)(6 9)(10 16)(11 17)(12 18)(13 19)(14 20)(15 21)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)
(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(16 20)(18 19)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 11)(2 10)(3 12)(4 20)(5 19)(6 21)(7 14)(8 13)(9 15)(16 24)(17 23)(18 22)

G:=sub<Sym(24)| (1,9)(2,7)(3,8)(4,24)(5,22)(6,23)(10,14)(11,15)(12,13)(16,20)(17,21)(18,19), (1,23)(2,24)(3,22)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,11)(2,12)(3,10)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (1,9)(2,7)(4,24)(6,23)(11,15)(12,13)(17,21)(18,19), (2,7)(3,8)(4,24)(5,22)(10,14)(12,13)(16,20)(18,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,11)(2,10)(3,12)(4,20)(5,19)(6,21)(7,14)(8,13)(9,15)(16,24)(17,23)(18,22)>;

G:=Group( (1,9)(2,7)(3,8)(4,24)(5,22)(6,23)(10,14)(11,15)(12,13)(16,20)(17,21)(18,19), (1,23)(2,24)(3,22)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,11)(2,12)(3,10)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (1,9)(2,7)(4,24)(6,23)(11,15)(12,13)(17,21)(18,19), (2,7)(3,8)(4,24)(5,22)(10,14)(12,13)(16,20)(18,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,11)(2,10)(3,12)(4,20)(5,19)(6,21)(7,14)(8,13)(9,15)(16,24)(17,23)(18,22) );

G=PermutationGroup([[(1,9),(2,7),(3,8),(4,24),(5,22),(6,23),(10,14),(11,15),(12,13),(16,20),(17,21),(18,19)], [(1,23),(2,24),(3,22),(4,7),(5,8),(6,9),(10,16),(11,17),(12,18),(13,19),(14,20),(15,21)], [(1,11),(2,12),(3,10),(4,19),(5,20),(6,21),(7,13),(8,14),(9,15),(16,22),(17,23),(18,24)], [(1,9),(2,7),(4,24),(6,23),(11,15),(12,13),(17,21),(18,19)], [(2,7),(3,8),(4,24),(5,22),(10,14),(12,13),(16,20),(18,19)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,11),(2,10),(3,12),(4,20),(5,19),(6,21),(7,14),(8,13),(9,15),(16,24),(17,23),(18,22)]])

G:=TransitiveGroup(24,400);

40 conjugacy classes

class 1 2A···2G2H···2O2P···2W 3 4A···4H6A···6G
order12···22···22···234···46···6
size11···13···36···686···68···8

40 irreducible representations

dim1112233
type+++++++
imageC1C2C2S3D6S4C2xS4
kernelC23xS4C22xS4C23xA4C25C24C23C22
# reps114117214

Matrix representation of C23xS4 in GL7(Z)

-1000000
0-100000
0010000
0001000
0000-100
00000-10
000000-1
,
-1000000
0-100000
0010000
0001000
0000100
0000010
0000001
,
-1000000
0-100000
00-10000
000-1000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000-100
00000-10
0000001
,
1000000
0100000
0010000
0001000
0000100
00000-10
000000-1
,
-1-100000
1000000
00-1-1000
0010000
0000001
0000100
0000010
,
-1000000
1100000
00-10000
0011000
0000-100
000000-1
00000-10

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],[-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],[-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],[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],[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],[-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0] >;

C23xS4 in GAP, Magma, Sage, TeX

C_2^3\times S_4
% in TeX

G:=Group("C2^3xS4");
// GroupNames label

G:=SmallGroup(192,1537);
// by ID

G=gap.SmallGroup(192,1537);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,1124,4037,285,2358,475]);
// Polycyclic

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

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