Copied to
clipboard

## G = D6.3S32order 432 = 24·33

### 3rd 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 — D6.3S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — D6.3S32
 Lower central C33 — C32×C6 — D6.3S32
 Upper central C1 — C2

Generators and relations for D6.3S32
G = < a,b,c,d,e,f | a3=b2=c6=e3=f2=1, d2=c3, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=bc3, dcd-1=c-1, ce=ec, cf=fc, de=ed, fdf=c3d, fef=e-1 >

Subgroups: 1436 in 218 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2 [×3], C3 [×3], C3 [×4], C4 [×4], C22 [×3], S3 [×13], C6 [×3], C6 [×8], C2×C4 [×3], D4 [×3], Q8, C32 [×3], C32 [×4], Dic3 [×2], Dic3 [×6], C12 [×8], D6, D6 [×10], C2×C6 [×4], C4○D4, C3×S3 [×6], C3⋊S3 [×10], C3×C6 [×3], C3×C6 [×5], Dic6 [×2], C4×S3 [×7], D12 [×5], C2×Dic3 [×2], C3⋊D4 [×5], C2×C12, C3×D4, C3×Q8, C33, C3×Dic3 [×4], C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×7], C62, C4○D12, D42S3, Q83S3, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, S3×Dic3 [×2], C6.D6 [×4], C3⋊D12, C3⋊D12 [×6], C322Q8, C3×Dic6 [×2], S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C327D4, C32×Dic3 [×2], C3×C3⋊Dic3 [×2], S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D12⋊S3, D6.6D6, D6.3D6, C3×S3×Dic3, C3×C3⋊D12, C3×C322Q8, C338(C2×C4), C337D4, C338D4, C339(C2×C4), D6.3S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12, D42S3, Q83S3, C2×S32 [×3], D12⋊S3, D6.6D6, D6.3D6, S33, D6.3S32

Permutation representations of D6.3S32
On 24 points - transitive group 24T1301
Generators in S24
```(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 18)(8 13)(9 14)(10 15)(11 16)(12 17)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 16 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)```

`G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23)>;`

`G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,16,4,13)(2,15,5,18)(3,14,6,17)(7,20,10,23)(8,19,11,22)(9,24,12,21), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23) );`

`G=PermutationGroup([(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,18),(8,13),(9,14),(10,15),(11,16),(12,17)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,16,4,13),(2,15,5,18),(3,14,6,17),(7,20,10,23),(8,19,11,22),(9,24,12,21)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,24),(8,19),(9,20),(10,21),(11,22),(12,23)])`

`G:=TransitiveGroup(24,1301);`

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 6K 6L 6M 12A 12B 12C ··· 12I 12J 12K 12L 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 6 12 12 12 ··· 12 12 12 12 size 1 1 6 18 54 2 2 2 4 4 4 8 6 6 9 9 18 2 2 2 4 4 4 6 6 8 12 12 12 36 6 6 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 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 Q8⋊3S3 C2×S32 D12⋊S3 D6.6D6 D6.3D6 S33 D6.3S32 kernel D6.3S32 C3×S3×Dic3 C3×C3⋊D12 C3×C32⋊2Q8 C33⋊8(C2×C4) C33⋊7D4 C33⋊8D4 C33⋊9(C2×C4) S3×Dic3 C3⋊D12 C32⋊2Q8 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C33 C32 Dic3 D6 C32 C32 C6 C3 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 4 2 2 1 2 4 2 1 1 1 3 2 2 2 1 1

Matrix representation of D6.3S32 in GL8(ℤ)

 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0
,
 0 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 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 0 0 0 0 0 0 0
,
 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 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
,
 -1 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 1 -1 0 0 0 0 0 0 0 0 -1 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 1 -1
,
 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 -1 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 -1 0 0 0 0 0

`G:=sub<GL(8,Integers())| [-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0],[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],[-1,-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,-1,-1,0,0,0,0,0,0,0,0,-1,-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,-1,-1],[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,-1,0,0,-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,-1,0,0,0,0,0] >;`

D6.3S32 in GAP, Magma, Sage, TeX

`D_6._3S_3^2`
`% in TeX`

`G:=Group("D6.3S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,609);`
`// by ID`

`G=gap.SmallGroup(432,609);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,298,2028,14118]);`
`// Polycyclic`

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

׿
×
𝔽