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G = S3xC6.D4  order 288 = 25·32

Direct product of S3 and C6.D4

direct product, metabelian, supersoluble, monomial

Aliases: S3xC6.D4, C62.110C23, C62:5(C2xC4), C23.24S32, D6:6(C2xDic3), (S3xC6).39D4, C6.166(S3xD4), D6:Dic3:27C2, C62:5C4:6C2, (C2xDic3):11D6, (S3xC23).3S3, C22:5(S3xDic3), D6.23(C3:D4), (C22xS3):4Dic3, (C22xS3).77D6, (C22xC6).117D6, (C6xDic3):13C22, (C2xC62).29C22, C6.18(C22xDic3), (S3xC2xC6):4C4, C6.97(S3xC2xC4), (C2xC6):17(C4xS3), C3:5(S3xC22:C4), C2.6(S3xC3:D4), (S3xC6):20(C2xC4), C22.54(C2xS32), (C2xS3xDic3):18C2, (C2xC6):4(C2xDic3), C6.62(C2xC3:D4), (S3xC22xC6).2C2, C32:7(C2xC22:C4), C2.18(C2xS3xDic3), (C3xC6).156(C2xD4), C3:1(C2xC6.D4), (S3xC2xC6).84C22, (C3xS3):2(C22:C4), (C3xC6).68(C22xC4), (C2xC3:Dic3):3C22, (C3xC6.D4):18C2, (C2xC6).129(C22xS3), SmallGroup(288,616)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — S3xC6.D4
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xS3xDic3 — S3xC6.D4
Lower central
C32 — C3xC6 — S3xC6.D4
Upper central
C1 — C22 — C23

Generators and relations for S3xC6.D4
 G = < a,b,c,d,e | a3=b2=c6=d4=1, e2=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >

Subgroups: 922 in 281 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2xC4, C23, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22xC4, C24, C3xS3, C3xS3, C3xC6, C3xC6, C3xC6, C4xS3, C2xDic3, C2xDic3, C2xC12, C22xS3, C22xS3, C22xS3, C22xC6, C22xC6, C2xC22:C4, C3xDic3, C3:Dic3, S3xC6, S3xC6, C62, C62, C62, D6:C4, C6.D4, C6.D4, C3xC22:C4, S3xC2xC4, C22xDic3, S3xC23, C23xC6, S3xDic3, C6xDic3, C2xC3:Dic3, S3xC2xC6, S3xC2xC6, S3xC2xC6, C2xC62, S3xC22:C4, C2xC6.D4, D6:Dic3, C3xC6.D4, C62:5C4, C2xS3xDic3, S3xC22xC6, S3xC6.D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, Dic3, D6, C22:C4, C22xC4, C2xD4, C4xS3, C2xDic3, C3:D4, C22xS3, C2xC22:C4, S32, C6.D4, S3xC2xC4, S3xD4, C22xDic3, C2xC3:D4, S3xDic3, C2xS32, S3xC22:C4, C2xC6.D4, C2xS3xDic3, S3xC3:D4, S3xC6.D4

Smallest permutation representation of S3xC6.D4
►On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)

(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 34)(14 35)(15 36)(16 31)(17 32)(18 33)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)

(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)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 23 16 30)(2 22 17 29)(3 21 18 28)(4 20 13 27)(5 19 14 26)(6 24 15 25)(7 42 44 34)(8 41 45 33)(9 40 46 32)(10 39 47 31)(11 38 48 36)(12 37 43 35)

(1 20 4 23)(2 19 5 22)(3 24 6 21)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 30 16 27)(14 29 17 26)(15 28 18 25)(37 46 40 43)(38 45 41 48)(39 44 42 47)
 
G:=sub<Sym(48)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,47,45)(44,48,46), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,23,16,30)(2,22,17,29)(3,21,18,28)(4,20,13,27)(5,19,14,26)(6,24,15,25)(7,42,44,34)(8,41,45,33)(9,40,46,32)(10,39,47,31)(11,38,48,36)(12,37,43,35), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,30,16,27)(14,29,17,26)(15,28,18,25)(37,46,40,43)(38,45,41,48)(39,44,42,47)>;
 
G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,47,45)(44,48,46), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,23,16,30)(2,22,17,29)(3,21,18,28)(4,20,13,27)(5,19,14,26)(6,24,15,25)(7,42,44,34)(8,41,45,33)(9,40,46,32)(10,39,47,31)(11,38,48,36)(12,37,43,35), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,30,16,27)(14,29,17,26)(15,28,18,25)(37,46,40,43)(38,45,41,48)(39,44,42,47) );
 
G=PermutationGroup([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,47,45),(44,48,46)], [(1,39),(2,40),(3,41),(4,42),(5,37),(6,38),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,34),(14,35),(15,36),(16,31),(17,32),(18,33),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,23,16,30),(2,22,17,29),(3,21,18,28),(4,20,13,27),(5,19,14,26),(6,24,15,25),(7,42,44,34),(8,41,45,33),(9,40,46,32),(10,39,47,31),(11,38,48,36),(12,37,43,35)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,34,10,31),(8,33,11,36),(9,32,12,35),(13,30,16,27),(14,29,17,26),(15,28,18,25),(37,46,40,43),(38,45,41,48),(39,44,42,47)]])
 

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C4A4B4C4D4E4F4G4H6A···6J6K···6S6T···6AA12A12B12C12D
order122222222222333444444446···66···66···612121212
size1111223333662246666181818182···24···46···612121212

54 irreducible representations

dim111111122222222244444
type++++++++++-++++-+
imageC1C2C2C2C2C2C4S3S3D4D6Dic3D6D6C3:D4C4xS3S32S3xD4S3xDic3C2xS32S3xC3:D4
kernelS3xC6.D4D6:Dic3C3xC6.D4C62:5C4C2xS3xDic3S3xC22xC6S3xC2xC6C6.D4S3xC23S3xC6C2xDic3C22xS3C22xS3C22xC6D6C2xC6C23C6C22C22C2
# reps121121811424228412214

Matrix representation of S3xC6.D4 ►in GL8(F13)

121000000
120000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
001200000
000120000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012000
000001200
000000012
000000112
,
120000000
012000000
008110000
00050000
000011000
000001200
000000012
000000120
,
120000000
012000000
00520000
00180000
000011000
000051200
00000001
00000010

G:=sub<GL(8,GF(13))| [12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

S3xC6.D4 in GAP, Magma, Sage, TeX 

S_3\times C_6.D_4
 
% in TeX
 
G:=Group("S3xC6.D4");
 
// GroupNames label
 
G:=SmallGroup(288,616);
 
// by ID
 
G=gap.SmallGroup(288,616);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^6=d^4=1,e^2=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^-1=c^-1,e*d*e^-1=c^3*d^-1>;
 
// generators/relations
 

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