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G = C22×C3⋊S4order 288 = 25·32

Direct product of C22 and C3⋊S4

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

Aliases: C22×C3⋊S4, C62(C2×S4), (C2×C6)⋊5S4, (C2×A4)⋊2D6, C32(C22×S4), (C23×C6)⋊5S3, (C22×C6)⋊3D6, C243(C3⋊S3), A42(C22×S3), (C22×A4)⋊5S3, (C6×A4)⋊3C22, (C3×A4)⋊3C23, (A4×C2×C6)⋊6C2, C23⋊(C2×C3⋊S3), C22⋊(C22×C3⋊S3), (C2×C6)⋊3(C22×S3), SmallGroup(288,1034)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C22×C3⋊S4
C1C22C2×C6C3×A4C3⋊S4C2×C3⋊S4 — C22×C3⋊S4
C3×A4 — C22×C3⋊S4
C1C22

Generators and relations for C22×C3⋊S4
 G = < a,b,c,d,e,f,g | a2=b2=c3=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, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 1876 in 326 conjugacy classes, 51 normal (11 characteristic)
C1, C2 [×3], C2 [×8], C3, C3 [×3], C4 [×4], C22 [×2], C22 [×26], S3 [×16], C6 [×3], C6 [×13], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], C32, Dic3 [×4], A4 [×3], D6 [×34], C2×C6 [×2], C2×C6 [×13], C22×C4, C2×D4 [×12], C24, C24, C3⋊S3 [×4], C3×C6 [×3], C2×Dic3 [×6], C3⋊D4 [×16], S4 [×12], C2×A4 [×9], C22×S3 [×13], C22×C6 [×3], C22×C6 [×4], C22×D4, C3×A4, C2×C3⋊S3 [×6], C62, C22×Dic3, C2×C3⋊D4 [×12], C2×S4 [×18], C22×A4 [×3], S3×C23, C23×C6, C3⋊S4 [×4], C6×A4 [×3], C22×C3⋊S3, C22×C3⋊D4, C22×S4 [×3], C2×C3⋊S4 [×6], A4×C2×C6, C22×C3⋊S4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C3⋊S3, S4, C22×S3 [×4], C2×C3⋊S3 [×3], C2×S4 [×3], C3⋊S4, C22×C3⋊S3, C22×S4, C2×C3⋊S4 [×3], C22×C3⋊S4

Smallest permutation representation of C22×C3⋊S4
On 36 points
Generators in S36
(1 25)(2 26)(3 27)(4 16)(5 17)(6 18)(7 22)(8 23)(9 24)(10 31)(11 32)(12 33)(13 28)(14 29)(15 30)(19 34)(20 35)(21 36)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)(22 25)(23 26)(24 27)(28 31)(29 32)(30 33)
(4 34)(5 35)(6 36)(10 13)(11 14)(12 15)(16 19)(17 20)(18 21)(28 31)(29 32)(30 33)
(1 10 16)(2 11 17)(3 12 18)(4 25 31)(5 26 32)(6 27 33)(7 13 19)(8 14 20)(9 15 21)(22 28 34)(23 29 35)(24 30 36)
(1 7)(2 9)(3 8)(4 28)(5 30)(6 29)(10 19)(11 21)(12 20)(13 16)(14 18)(15 17)(22 25)(23 27)(24 26)(31 34)(32 36)(33 35)

G:=sub<Sym(36)| (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35)>;

G:=Group( (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35) );

G=PermutationGroup([(1,25),(2,26),(3,27),(4,16),(5,17),(6,18),(7,22),(8,23),(9,24),(10,31),(11,32),(12,33),(13,28),(14,29),(15,30),(19,34),(20,35),(21,36)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,7),(2,8),(3,9),(10,13),(11,14),(12,15),(22,25),(23,26),(24,27),(28,31),(29,32),(30,33)], [(4,34),(5,35),(6,36),(10,13),(11,14),(12,15),(16,19),(17,20),(18,21),(28,31),(29,32),(30,33)], [(1,10,16),(2,11,17),(3,12,18),(4,25,31),(5,26,32),(6,27,33),(7,13,19),(8,14,20),(9,15,21),(22,28,34),(23,29,35),(24,30,36)], [(1,7),(2,9),(3,8),(4,28),(5,30),(6,29),(10,19),(11,21),(12,20),(13,16),(14,18),(15,17),(22,25),(23,27),(24,26),(31,34),(32,36),(33,35)])

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C3D4A4B4C4D6A6B6C6D6E6F6G6H···6P
order1222222222223333444466666666···6
size111133331818181828881818181822266668···8

36 irreducible representations

dim11122223366
type+++++++++++
imageC1C2C2S3S3D6D6S4C2×S4C3⋊S4C2×C3⋊S4
kernelC22×C3⋊S4C2×C3⋊S4A4×C2×C6C22×A4C23×C6C2×A4C22×C6C2×C6C6C22C2
# reps16131932613

Matrix representation of C22×C3⋊S4 in GL7(ℤ)

-1000000
0-100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
00-10000
000-1000
0000100
0000010
0000001
,
-1100000
-1000000
00-11000
00-10000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000100
0000-1-10
000010-1
,
1000000
0100000
0010000
0001000
0000-100
00000-10
0000-101
,
1000000
0100000
0010000
0001000
0000120
00000-1-1
0000010
,
0100000
1000000
0001000
0010000
0000-1-20
0000010
00000-1-1

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,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[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,-1,1,0,0,0,0,0,-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,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,-1,1,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-2,1,-1,0,0,0,0,0,0,-1] >;

C22×C3⋊S4 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes S_4
% in TeX

G:=Group("C2^2xC3:S4");
// GroupNames label

G:=SmallGroup(288,1034);
// by ID

G=gap.SmallGroup(288,1034);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,451,1684,6053,782,3534,1350]);
// Polycyclic

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