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G = D6⋊S3⋊S3order 432 = 24·33

5th semidirect product of D6⋊S3 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — D6⋊S3⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×D6⋊S3 — D6⋊S3⋊S3
 Lower central C33 — C32×C6 — D6⋊S3⋊S3
 Upper central C1 — C2

Generators and relations for D6⋊S3⋊S3
G = < a,b,c,d,e,f | a6=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=fbf=a3b, be=eb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=a3d, fef=e-1 >

Subgroups: 1076 in 214 conjugacy classes, 46 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, D42S3, S3×C32, C32×C6, S3×Dic3, D6⋊S3, C322Q8, C3×C3⋊D4, C2×C3⋊Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, D6.4D6, C3×D6⋊S3, S3×C3⋊Dic3, C335Q8, D6⋊S3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D42S3, C2×S32, D6.4D6, S33, D6⋊S3⋊S3

Smallest permutation representation of D6⋊S3⋊S3
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)(25 31)(26 36)(27 35)(28 34)(29 33)(30 32)(37 46)(38 45)(39 44)(40 43)(41 48)(42 47)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 47 45)(44 48 46)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H ··· 6S 12A 12B 12C order 1 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 ··· 6 12 12 12 size 1 1 6 6 6 2 2 2 4 4 4 8 18 18 18 27 27 2 2 2 4 4 4 8 12 ··· 12 36 36 36

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 4 4 4 8 8 type + + + + + + + + - + - + - image C1 C2 C2 C2 S3 D6 D6 C4○D4 S32 D4⋊2S3 C2×S32 D6.4D6 S33 D6⋊S3⋊S3 kernel D6⋊S3⋊S3 C3×D6⋊S3 S3×C3⋊Dic3 C33⋊5Q8 D6⋊S3 C3⋊Dic3 S3×C6 C33 D6 C32 C6 C3 C2 C1 # reps 1 3 3 1 3 3 6 2 3 3 3 6 1 1

Matrix representation of D6⋊S3⋊S3 in GL8(𝔽13)

 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 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 0 0 12 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 12 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 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
,
 2 9 0 0 0 0 0 0 4 11 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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
,
 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 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 7 3 0 0 0 0 0 0 10 6 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 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 1 0

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[2,4,0,0,0,0,0,0,9,11,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

D6⋊S3⋊S3 in GAP, Magma, Sage, TeX

`D_6\rtimes S_3\rtimes S_3`
`% in TeX`

`G:=Group("D6:S3:S3");`
`// GroupNames label`

`G:=SmallGroup(432,610);`
`// by ID`

`G=gap.SmallGroup(432,610);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,254,135,298,2028,14118]);`
`// Polycyclic`

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

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