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G = S3xC6.D6  order 432 = 24·33

Direct product of S3 and C6.D6

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

Aliases: S3xC6.D6, D6.8S32, Dic3:5S32, C3:Dic3:12D6, (S3xDic3):6S3, (C3xDic3):8D6, (S3xC6).18D6, C33:3(C22xC4), (C32xC6).2C23, (C32xDic3):10C22, C2.2S33, C3:1(C4xS32), C6.2(C2xS32), (S3xC3:S3):1C4, C3:S3:4(C4xS3), C32:6(S3xC2xC4), (C3xS3):1(C4xS3), (C2xC3:S3).28D6, (C3xS3xDic3):10C2, C3:1(C2xC6.D6), C33:9(C2xC4):10C2, C33:8(C2xC4):10C2, (S3xC3xC6).2C22, (S3xC32):3(C2xC4), C33:C2:2(C2xC4), (C3xC6.D6):9C2, (C6xC3:S3).15C22, (C3xC6).51(C22xS3), (C3xC3:Dic3):9C22, (C2xC33:C2).1C22, (C2xS3xC3:S3).2C2, (C3xC3:S3):2(C2xC4), SmallGroup(432,595)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — S3xC6.D6
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C3xS3xDic3 — S3xC6.D6
Lower central
C33 — S3xC6.D6
Upper central
C1 — C2

Generators and relations for S3xC6.D6
 G = < a,b,c,d,e | a3=b2=c6=e2=1, d6=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1788 in 290 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, S3, C6, C6, C6, C2xC4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C22xC4, C3xS3, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xC2xC4, S3xC32, C3xC3:S3, C33:C2, C32xC6, S3xDic3, S3xDic3, C6.D6, C6.D6, S3xC12, C6xDic3, C4xC3:S3, C2xS32, C22xC3:S3, C32xDic3, C3xC3:Dic3, S3xC3:S3, S3xC3xC6, C6xC3:S3, C2xC33:C2, C4xS32, C2xC6.D6, C3xS3xDic3, C3xC6.D6, C33:8(C2xC4), C33:9(C2xC4), C2xS3xC3:S3, S3xC6.D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S32, S3xC2xC4, C6.D6, C2xS32, C4xS32, C2xC6.D6, S33, S3xC6.D6

Permutation representations of S3xC6.D6
►On 24 points - transitive group 24T1298
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)

(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)

(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)

(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 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)
 
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22)>;
 
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22) );
 
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15),(2,20),(3,13),(4,18),(5,23),(6,16),(7,21),(8,14),(9,19),(10,24),(11,17),(12,22)]])
 
G:=TransitiveGroup(24,1298);
 

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O12A···12H12I12J12K12L12M12N12O12P
order1222222233333334444444466666666666666612···121212121212121212
size113399272722244483333999922244466668121218186···61212121218181818

54 irreducible representations

dim1111111222222224444488
type++++++++++++++++++
imageC1C2C2C2C2C2C4S3S3D6D6D6D6C4xS3C4xS3S32S32C6.D6C2xS32C4xS32S33S3xC6.D6
kernelS3xC6.D6C3xS3xDic3C3xC6.D6C33:8(C2xC4)C33:9(C2xC4)C2xS3xC3:S3S3xC3:S3S3xDic3C6.D6C3xDic3C3:Dic3S3xC6C2xC3:S3C3xS3C3:S3Dic3D6S3C6C3C2C1
# reps1212118214221842123411

Matrix representation of S3xC6.D6 ►in GL6(F13)

1210000
1200000
001000
000100
000010
000001
,
0120000
1200000
0012000
0001200
000010
000001
,
100000
010000
000100
0012100
000010
000001
,
1200000
0120000
000500
005000
0000012
0000112
,
1200000
0120000
0001200
0012000
0000112
0000012

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
 

S3xC6.D6 in GAP, Magma, Sage, TeX 

S_3\times C_6.D_6
 
% in TeX
 
G:=Group("S3xC6.D6");
 
// GroupNames label
 
G:=SmallGroup(432,595);
 
// by ID
 
G=gap.SmallGroup(432,595);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^6=e^2=1,d^6=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
 
// generators/relations
 

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