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## G = C23×S4order 192 = 26·3

### Direct product of C23 and S4

Aliases: C23×S4, A4⋊C24, C252S3, C244D6, (C2×A4)⋊C23, C22⋊(S3×C23), C23⋊(C22×S3), (C23×A4)⋊3C2, (C22×A4)⋊4C22, SmallGroup(192,1537)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C23×S4
 Chief series C1 — C22 — A4 — S4 — C2×S4 — C22×S4 — C23×S4
 Lower central A4 — C23×S4
 Upper central C1 — C23

Generators and relations for C23×S4
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 [×7], C2 [×16], C3, C4 [×8], C22, C22 [×7], C22 [×106], S3 [×8], C6 [×7], C2×C4 [×28], D4 [×64], C23, C23 [×7], C23 [×126], A4, D6 [×28], C2×C6 [×7], C22×C4 [×14], C2×D4 [×112], C24 [×7], C24 [×30], S4 [×8], C2×A4 [×7], C22×S3 [×14], C22×C6, C23×C4, C22×D4 [×28], C25, C25, C2×S4 [×28], C22×A4 [×7], S3×C23, D4×C23, C22×S4 [×14], C23×A4, C23×S4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, S4, C22×S3 [×7], C2×S4 [×7], S3×C23, C22×S4 [×7], C23×S4

Permutation representations of C23×S4
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 ··· 2G 2H ··· 2O 2P ··· 2W 3 4A ··· 4H 6A ··· 6G order 1 2 ··· 2 2 ··· 2 2 ··· 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 3 ··· 3 6 ··· 6 8 6 ··· 6 8 ··· 8

40 irreducible representations

 dim 1 1 1 2 2 3 3 type + + + + + + + image C1 C2 C2 S3 D6 S4 C2×S4 kernel C23×S4 C22×S4 C23×A4 C25 C24 C23 C22 # reps 1 14 1 1 7 2 14

Matrix representation of C23×S4 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
,
 -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 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 0 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 -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],[-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] >;

C23×S4 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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