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G = S3×C6.D4  order 288 = 25·32

Direct product of S3 and C6.D4

direct product, metabelian, supersoluble, monomial

Aliases: S3×C6.D4, C62.110C23, C62⋊5(C2×C4), C23.24S32, D6⋊6(C2×Dic3), (S3×C6).39D4, C6.166(S3×D4), D6⋊Dic3⋊27C2, C62⋊5C4⋊6C2, (C2×Dic3)⋊11D6, (S3×C23).3S3, C22⋊5(S3×Dic3), D6.23(C3⋊D4), (C22×S3)⋊4Dic3, (C22×S3).77D6, (C22×C6).117D6, (C6×Dic3)⋊13C22, (C2×C62).29C22, C6.18(C22×Dic3), (S3×C2×C6)⋊4C4, C6.97(S3×C2×C4), (C2×C6)⋊17(C4×S3), C3⋊5(S3×C22⋊C4), C2.6(S3×C3⋊D4), (S3×C6)⋊20(C2×C4), C22.54(C2×S32), (C2×S3×Dic3)⋊18C2, (C2×C6)⋊4(C2×Dic3), C6.62(C2×C3⋊D4), (S3×C22×C6).2C2, C32⋊7(C2×C22⋊C4), C2.18(C2×S3×Dic3), (C3×C6).156(C2×D4), C3⋊1(C2×C6.D4), (S3×C2×C6).84C22, (C3×S3)⋊2(C22⋊C4), (C3×C6).68(C22×C4), (C2×C3⋊Dic3)⋊3C22, (C3×C6.D4)⋊18C2, (C2×C6).129(C22×S3), SmallGroup(288,616)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — S3×C6.D4
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — S3×C6.D4
Lower central
C32 — C3×C6 — S3×C6.D4
Upper central
C1 — C22 — C23

Generators and relations for S3×C6.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, C2×C4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, S3×C23, C23×C6, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, S3×C2×C6, S3×C2×C6, C2×C62, S3×C22⋊C4, C2×C6.D4, D6⋊Dic3, C3×C6.D4, C62⋊5C4, C2×S3×Dic3, S3×C22×C6, S3×C6.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, S32, C6.D4, S3×C2×C4, S3×D4, C22×Dic3, C2×C3⋊D4, S3×Dic3, C2×S32, S3×C22⋊C4, C2×C6.D4, C2×S3×Dic3, S3×C3⋊D4, S3×C6.D4

Smallest permutation representation of S3×C6.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⋊D4C4×S3S32S3×D4S3×Dic3C2×S32S3×C3⋊D4
kernelS3×C6.D4D6⋊Dic3C3×C6.D4C62⋊5C4C2×S3×Dic3S3×C22×C6S3×C2×C6C6.D4S3×C23S3×C6C2×Dic3C22×S3C22×S3C22×C6D6C2×C6C23C6C22C22C2
# reps121121811424228412214

Matrix representation of S3×C6.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] >;
 

S3×C6.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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