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## G = D6.4S32order 432 = 24·33

### 1st non-split extension by D6 of S32 acting via S32/C3⋊S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — D6.4S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — D6.4S32
 Lower central C33 — C32×C6 — D6.4S32
 Upper central C1 — C2

Generators and relations for D6.4S32
G = < a,b,c,d,e | a3=b12=c2=d3=e2=1, bab-1=cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1116 in 210 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×C32, C3×C3⋊S3, C32×C6, S3×Dic3, S3×Dic3, C6.D6, D6⋊S3, D6⋊S3, C3⋊D12, C322Q8, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C324Q8, C2×C3⋊Dic3, C327D4, C32×Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, C6×C3⋊S3, D125S3, D6.3D6, D6.4D6, C3×S3×Dic3, C3×D6⋊S3, C3×C3⋊D12, S3×C3⋊Dic3, C336D4, C334Q8, C339(C2×C4), D6.4S32
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D42S3, C2×S32, D125S3, D6.3D6, D6.4D6, S33, D6.4S32

Smallest permutation representation of D6.4S32
On 48 points
Generators in S48
```(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 41 45)(38 46 42)(39 43 47)(40 48 44)
(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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)```

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

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

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

42 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 6I 6J ··· 6P 6Q 12A 12B 12C 12D 12E 12F 12G 12H 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 6 ··· 6 6 12 12 12 12 12 12 12 12 size 1 1 6 6 18 2 2 2 4 4 4 8 6 9 9 18 54 2 2 2 4 4 4 6 6 8 12 ··· 12 36 6 6 12 12 12 18 18 36

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + - + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 S3 D6 D6 D6 D6 C4○D4 C4○D12 S32 S32 D4⋊2S3 C2×S32 D12⋊5S3 D6.3D6 D6.4D6 S33 D6.4S32 kernel D6.4S32 C3×S3×Dic3 C3×D6⋊S3 C3×C3⋊D12 S3×C3⋊Dic3 C33⋊6D4 C33⋊4Q8 C33⋊9(C2×C4) S3×Dic3 D6⋊S3 C3⋊D12 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C33 C32 Dic3 D6 C32 C6 C3 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 2 4 1 2 4 1 2 2 3 2 2 2 1 1

Matrix representation of D6.4S32 in GL8(𝔽13)

 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 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 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1
,
 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 12 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 1 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 0 1 0 0 0 0 0 0 12 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
,
 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 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 12 0 0 0 0 0 0 0 0 12

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

D6.4S32 in GAP, Magma, Sage, TeX

`D_6._4S_3^2`
`% in TeX`

`G:=Group("D6.4S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,608);`
`// by ID`

`G=gap.SmallGroup(432,608);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,298,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^3=e^2=1,b*a*b^-1=c*a*c=a^-1,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,e*c*e=b^6*c,e*d*e=d^-1>;`
`// generators/relations`

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