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G = C22xS3xA4order 288 = 25·32

Direct product of C22, S3 and A4

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

Aliases: C22xS3xA4, C6:(C22xA4), C3:(C23xA4), C23:4(S3xC6), (C23xC6):5C6, C24:8(C3xS3), (S3xC24):1C3, (C3xA4):4C23, (S3xC23):3C6, (C6xA4):4C22, (A4xC2xC6):7C2, (C2xC6):5(C2xA4), (C2xC6):(C22xC6), (C22xC6):(C2xC6), C22:2(S3xC2xC6), (C22xS3):5(C2xC6), SmallGroup(288,1037)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C22xS3xA4
C1C3C2xC6C3xA4S3xA4C2xS3xA4 — C22xS3xA4
C2xC6 — C22xS3xA4
C1C22

Generators and relations for C22xS3xA4
 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, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C23, C32, A4, A4, D6, D6, C2xC6, C2xC6, C24, C24, C3xS3, C3xC6, C2xA4, C2xA4, C22xS3, C22xS3, C22xS3, C22xC6, C22xC6, C25, C3xA4, S3xC6, C62, C22xA4, C22xA4, S3xC23, S3xC23, C23xC6, S3xA4, C6xA4, S3xC2xC6, C23xA4, S3xC24, C2xS3xA4, A4xC2xC6, C22xS3xA4
Quotients: C1, C2, C3, C22, S3, C6, C23, A4, D6, C2xC6, C3xS3, C2xA4, C22xS3, C22xC6, S3xC6, C22xA4, S3xA4, S3xC2xC6, C23xA4, C2xS3xA4, C22xS3xA4

Smallest permutation representation of C22xS3xA4
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++++++++++
imageC1C2C2C3C6C6S3D6C3xS3S3xC6A4C2xA4C2xA4S3xA4C2xS3xA4
kernelC22xS3xA4C2xS3xA4A4xC2xC6S3xC24S3xC23C23xC6C22xA4C2xA4C24C23C22xS3D6C2xC6C22C2
# reps1612122132616113

Matrix representation of C22xS3xA4 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] >;

C22xS3xA4 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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