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G = C22×S3×A4order 288 = 25·32

Direct product of C22, S3 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: C22×S3×A4, C6⋊(C22×A4), C3⋊(C23×A4), C234(S3×C6), (C23×C6)⋊5C6, C248(C3×S3), (S3×C24)⋊1C3, (C3×A4)⋊4C23, (S3×C23)⋊3C6, (C6×A4)⋊4C22, (A4×C2×C6)⋊7C2, (C2×C6)⋊5(C2×A4), (C2×C6)⋊(C22×C6), (C22×C6)⋊(C2×C6), C222(S3×C2×C6), (C22×S3)⋊5(C2×C6), SmallGroup(288,1037)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C22×S3×A4
C1C3C2×C6C3×A4S3×A4C2×S3×A4 — C22×S3×A4
C2×C6 — C22×S3×A4
C1C22

Generators and relations for C22×S3×A4
 G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 1682 in 366 conjugacy classes, 63 normal (15 characteristic)
C1, C2 [×3], C2 [×12], C3, C3 [×2], C22 [×2], C22 [×55], S3 [×4], S3 [×4], C6 [×3], C6 [×14], C23 [×3], C23 [×54], C32, A4, A4, D6 [×6], D6 [×38], C2×C6 [×2], C2×C6 [×19], C24, C24 [×14], C3×S3 [×4], C3×C6 [×3], C2×A4 [×3], C2×A4 [×7], C22×S3, C22×S3 [×4], C22×S3 [×45], C22×C6 [×3], C22×C6 [×5], C25, C3×A4, S3×C6 [×6], C62, C22×A4, C22×A4 [×7], S3×C23 [×6], S3×C23 [×8], C23×C6, S3×A4 [×4], C6×A4 [×3], S3×C2×C6, C23×A4, S3×C24, C2×S3×A4 [×6], A4×C2×C6, C22×S3×A4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, A4, D6 [×3], C2×C6 [×7], C3×S3, C2×A4 [×7], C22×S3, C22×C6, S3×C6 [×3], C22×A4 [×7], S3×A4, S3×C2×C6, C23×A4, C2×S3×A4 [×3], C22×S3×A4

Smallest permutation representation of C22×S3×A4
On 36 points
Generators in S36
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(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 22)(2 24)(3 23)(4 19)(5 21)(6 20)(7 28)(8 30)(9 29)(10 25)(11 27)(12 26)(13 34)(14 36)(15 35)(16 31)(17 33)(18 32)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D···2K2L2M2N2O3A3B3C3D3E6A6B6C6D···6I6J6K6L6M6N···6S6T···6AA
order12222···22222333336666···666666···66···6
size11113···39999244882224···466668···812···12

48 irreducible representations

dim111111222233366
type++++++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6A4C2×A4C2×A4S3×A4C2×S3×A4
kernelC22×S3×A4C2×S3×A4A4×C2×C6S3×C24S3×C23C23×C6C22×A4C2×A4C24C23C22×S3D6C2×C6C22C2
# reps1612122132616113

Matrix representation of C22×S3×A4 in GL7(𝔽7)

6000000
0600000
0060000
0006000
0000100
0000010
0000001
,
6000000
0600000
0010000
0001000
0000100
0000010
0000001
,
0600000
1600000
0066000
0010000
0000100
0000010
0000001
,
1600000
0600000
0060000
0011000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000600
0000010
0000006
,
1000000
0100000
0010000
0001000
0000600
0000060
0000001
,
2000000
0200000
0010000
0001000
0000001
0000600
0000060

G:=sub<GL(7,GF(7))| [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,1,0,0] >;

C22×S3×A4 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times A_4
% in TeX

G:=Group("C2^2xS3xA4");
// GroupNames label

G:=SmallGroup(288,1037);
// by ID

G=gap.SmallGroup(288,1037);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-3,340,152,9414]);
// Polycyclic

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

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