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## G = C12.86S32order 432 = 24·33

### 6th non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C12.86S32
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C2×C32⋊D6 — C12.86S32
 Lower central He3 — C2×He3 — C12.86S32
 Upper central C1 — C2 — C4

Generators and relations for C12.86S32
G = < a,b,c,d | a3=b12=c6=d2=1, ab=ba, cac-1=dad=a-1b4, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 1555 in 205 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4, C4, C22 [×9], S3 [×14], C6, C6 [×9], C2×C4, D4 [×4], C23 [×2], C32 [×2], C32, Dic3 [×3], C12, C12 [×4], D6 [×23], C2×C6 [×5], C2×D4, C3×S3 [×10], C3⋊S3 [×4], C3×C6 [×2], C3×C6, C4×S3 [×3], D12 [×6], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×6], He3, C3×Dic3 [×3], C3×C12 [×2], C3×C12, S32 [×8], S3×C6 [×7], C2×C3⋊S3 [×4], C2×D12, S3×D4 [×2], C32⋊C6 [×4], He3⋊C2 [×2], C2×He3, C3⋊D12 [×4], S3×C12 [×3], C3×D12 [×2], C12⋊S3 [×2], C2×S32 [×4], He33C4, C4×He3, C32⋊D6 [×4], C2×C32⋊C6 [×4], C2×He3⋊C2, S3×D12 [×2], He32D4 [×2], He34D4 [×2], C4×He3⋊C2, C2×C32⋊D6 [×2], C12.86S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], S32, S3×D4 [×2], C2×S32, C32⋊D6, D6⋊D6, C2×C32⋊D6, C12.86S32

Smallest permutation representation of C12.86S32
On 36 points
Generators in S36
```(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(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)
(1 14 29)(2 13 30 12 15 28)(3 24 31 11 16 27)(4 23 32 10 17 26)(5 22 33 9 18 25)(6 21 34 8 19 36)(7 20 35)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)```

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

`G:=Group( (13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (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), (1,14,29)(2,13,30,12,15,28)(3,24,31,11,16,27)(4,23,32,10,17,26)(5,22,33,9,18,25)(6,21,34,8,19,36)(7,20,35), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28) );`

`G=PermutationGroup([(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36)], [(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)], [(1,14,29),(2,13,30,12,15,28),(3,24,31,11,16,27),(4,23,32,10,17,26),(5,22,33,9,18,25),(6,21,34,8,19,36),(7,20,35)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28)])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 9 9 18 18 18 18 2 6 6 12 2 18 2 6 6 12 18 18 36 36 36 36 2 2 12 12 12 12 18 18

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 6 6 6 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D6 S32 S3×D4 C2×S32 D6⋊D6 C32⋊D6 C2×C32⋊D6 C12.86S32 kernel C12.86S32 He3⋊2D4 He3⋊4D4 C4×He3⋊C2 C2×C32⋊D6 C12⋊S3 He3⋊C2 C3×C12 C2×C3⋊S3 C12 C32 C6 C3 C4 C2 C1 # reps 1 2 2 1 2 2 2 2 4 1 2 1 2 2 2 4

Matrix representation of C12.86S32 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 3 3 0 0 0 0 10 6 0 0 0 0 0 0 3 3 0 0 0 0 10 6 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 0 0 0 0 0 12 0 0 0 0 12 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 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 1 0 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;`

C12.86S32 in GAP, Magma, Sage, TeX

`C_{12}._{86}S_3^2`
`% in TeX`

`G:=Group("C12.86S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,302);`
`// by ID`

`G=gap.SmallGroup(432,302);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,571,4037,537,14118,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^12=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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