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G = C2xC6xS4order 288 = 25·32

Direct product of C2xC6 and S4

direct product, non-abelian, soluble, monomial

Aliases: C2xC6xS4, C23:(S3xC6), A4:(C22xC6), C24:4(C3xS3), (C23xC6):2S3, (C22xC6):2D6, (C6xA4):2C22, (C22xA4):5C6, (C3xA4):2C23, C22:(S3xC2xC6), (C2xA4):(C2xC6), (A4xC2xC6):5C2, (C2xC6):2(C22xS3), SmallGroup(288,1033)

Series: Derived Chief Lower central Upper central

C1C22A4 — C2xC6xS4
C1C22A4C3xA4C3xS4C6xS4 — C2xC6xS4
A4 — C2xC6xS4
C1C2xC6

Generators and relations for C2xC6xS4
 G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 890 in 272 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, C12, A4, A4, D6, C2xC6, C2xC6, C22xC4, C2xD4, C24, C24, C3xS3, C3xC6, C2xC12, C3xD4, S4, C2xA4, C2xA4, C22xS3, C22xC6, C22xC6, C22xD4, C3xA4, S3xC6, C62, C22xC12, C6xD4, C2xS4, C22xA4, C22xA4, C23xC6, C23xC6, C3xS4, C6xA4, S3xC2xC6, D4xC2xC6, C22xS4, C6xS4, A4xC2xC6, C2xC6xS4
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C3xS3, S4, C22xS3, C22xC6, S3xC6, C2xS4, C3xS4, S3xC2xC6, C22xS4, C6xS4, C2xC6xS4

Smallest permutation representation of C2xC6xS4
On 36 points
Generators in S36
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 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 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 12 32)(2 7 33)(3 8 34)(4 9 35)(5 10 36)(6 11 31)(13 27 19)(14 28 20)(15 29 21)(16 30 22)(17 25 23)(18 26 24)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C3D3E4A4B4C4D6A···6F6G···6N6O···6V6W···6AE12A···12H
order1222222222223333344446···66···66···66···612···12
size1111333366661188866661···13···36···68···86···6

60 irreducible representations

dim11111122223333
type+++++++
imageC1C2C2C3C6C6S3D6C3xS3S3xC6S4C2xS4C3xS4C6xS4
kernelC2xC6xS4C6xS4A4xC2xC6C22xS4C2xS4C22xA4C23xC6C22xC6C24C23C2xC6C6C22C2
# reps1612122132626412

Matrix representation of C2xC6xS4 in GL5(F13)

120000
012000
001200
000120
000012
,
40000
04000
00100
00010
00001
,
10000
01000
001200
000120
00111
,
10000
01000
00100
000120
0012012
,
012000
112000
00121211
00100
00001
,
121000
01000
001200
00112
000012

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,12,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,12,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,11,0,1],[12,0,0,0,0,1,1,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,2,12] >;

C2xC6xS4 in GAP, Magma, Sage, TeX

C_2\times C_6\times S_4
% in TeX

G:=Group("C2xC6xS4");
// GroupNames label

G:=SmallGroup(288,1033);
// by ID

G=gap.SmallGroup(288,1033);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,1684,6053,285,3534,475]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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