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G = C23×S4order 192 = 26·3

Direct product of C23 and S4

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

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

C1C22A4 — C23×S4
C1C22A4S4C2×S4C22×S4 — C23×S4
A4 — C23×S4
C1C23

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

40 irreducible representations

dim1112233
type+++++++
imageC1C2C2S3D6S4C2×S4
kernelC23×S4C22×S4C23×A4C25C24C23C22
# reps114117214

Matrix representation of C23×S4 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] >;

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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