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G = S3×C6×Dic3  order 432 = 24·33

Direct product of C6, S3 and Dic3

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

Aliases: S3×C6×Dic3, C62.108D6, C6⋊3(S3×C12), (S3×C6)⋊3C12, D6.9(S3×C6), C6⋊1(C6×Dic3), (C6×Dic3)⋊7C6, (S3×C6).48D6, (S3×C62).4C2, C62.22(C2×C6), C33⋊10(C22×C4), C32⋊5(C22×C12), (C3×C62).16C22, (C32×C6).29C23, C32⋊8(C22×Dic3), (C32×Dic3)⋊17C22, (S3×C3×C6)⋊6C4, C2.2(S32×C6), C3⋊4(S3×C2×C12), (C2×C6).71S32, (S3×C2×C6).8C6, C6.10(S3×C2×C6), C3⋊1(Dic3×C2×C6), C6.113(C2×S32), (C3×C6)⋊4(C2×C12), (C3×C6)⋊12(C4×S3), (S3×C2×C6).11S3, C32⋊19(S3×C2×C4), C22.8(C3×S32), (C3×S3)⋊2(C2×C12), (C2×C6).26(S3×C6), (C6×C3⋊Dic3)⋊7C2, C3⋊Dic3⋊8(C2×C6), (C32×C6)⋊4(C2×C4), (Dic3×C3×C6)⋊11C2, (C3×C6)⋊7(C2×Dic3), (S3×C6).15(C2×C6), (S3×C32)⋊6(C2×C4), (C2×C3⋊Dic3)⋊10C6, (C3×Dic3)⋊6(C2×C6), (S3×C3×C6).22C22, (C22×S3).2(C3×S3), (C3×C6).20(C22×C6), (C3×C6).134(C22×S3), (C3×C3⋊Dic3)⋊12C22, SmallGroup(432,651)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — S3×C6×Dic3
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — S3×C6×Dic3
Lower central
C32 — S3×C6×Dic3
Upper central
C1 — C2×C6

Generators and relations for S3×C6×Dic3
 G = < a,b,c,d,e | a6=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 768 in 290 conjugacy classes, 112 normal (36 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C62, S3×C2×C4, C22×Dic3, C22×C12, S3×C32, C32×C6, C32×C6, S3×Dic3, S3×C12, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, S3×C2×C6, C2×C62, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C3×C62, C2×S3×Dic3, S3×C2×C12, Dic3×C2×C6, C3×S3×Dic3, Dic3×C3×C6, C6×C3⋊Dic3, S3×C62, S3×C6×Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, S32, S3×C6, S3×C2×C4, C22×Dic3, C22×C12, S3×Dic3, S3×C12, C6×Dic3, C2×S32, S3×C2×C6, C3×S32, C2×S3×Dic3, S3×C2×C12, Dic3×C2×C6, C3×S3×Dic3, S32×C6, S3×C6×Dic3

Smallest permutation representation of S3×C6×Dic3
►On 48 points
Generators in S48
(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 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)

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

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

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

108 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H3I3J3K4A4B4C4D4E4F4G4H6A···6F6G···6X6Y···6AF6AG···6AO6AP···6BA12A···12H12I···12T12U···12AB
order12222222333···3333444444446···66···66···66···66···612···1212···1212···12
size11113333112···2444333399991···12···23···34···46···63···36···69···9

108 irreducible representations

dim11111111111122222222222222444444
type++++++++-+++-+
imageC1C2C2C2C2C3C4C6C6C6C6C12S3S3D6Dic3D6D6C3×S3C3×S3C4×S3S3×C6C3×Dic3S3×C6S3×C6S3×C12S32S3×Dic3C2×S32C3×S32C3×S3×Dic3S32×C6
kernelS3×C6×Dic3C3×S3×Dic3Dic3×C3×C6C6×C3⋊Dic3S3×C62C2×S3×Dic3S3×C3×C6S3×Dic3C6×Dic3C2×C3⋊Dic3S3×C2×C6S3×C6C6×Dic3S3×C2×C6C3×Dic3S3×C6S3×C6C62C2×Dic3C22×S3C3×C6Dic3D6D6C2×C6C6C2×C6C6C6C22C2C2
# reps141112882221611242222448448121242

Matrix representation of S3×C6×Dic3 ►in GL6(𝔽13)

1200000
0120000
003000
000300
0000120
0000012
,
010000
12120000
000100
00121200
000010
000001
,
1200000
110000
001000
00121200
0000120
0000012
,
100000
010000
001000
000100
000011
0000120
,
1200000
0120000
001000
000100
000080
000055

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,5,0,0,0,0,0,5] >;
 

S3×C6×Dic3 in GAP, Magma, Sage, TeX 

S_3\times C_6\times {\rm Dic}_3
 
% in TeX
 
G:=Group("S3xC6xDic3");
 
// GroupNames label
 
G:=SmallGroup(432,651);
 
// by ID
 
G=gap.SmallGroup(432,651);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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