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## G = C33⋊8D4order 216 = 23·33

### 5th semidirect product of C33 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊8D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C33⋊8D4
 Lower central C33 — C32×C6 — C33⋊8D4
 Upper central C1 — C2

Generators and relations for C338D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 852 in 136 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×17], C6, C6 [×4], C6 [×5], D4, C32, C32 [×4], C32 [×4], Dic3, C12 [×4], D6 [×13], C2×C6, C3×S3 [×4], C3⋊S3 [×14], C3×C6, C3×C6 [×4], C3×C6 [×4], D12 [×4], C3⋊D4, C33, C3×Dic3 [×4], C3×C12, S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×9], C3×C3⋊S3, C33⋊C2, C32×C6, C3⋊D12 [×4], C12⋊S3, C32×Dic3, C6×C3⋊S3, C2×C33⋊C2, C338D4
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C338D4

Smallest permutation representation of C338D4
On 36 points
Generators in S36
```(1 29 8)(2 30 5)(3 31 6)(4 32 7)(9 33 27)(10 34 28)(11 35 25)(12 36 26)(13 22 18)(14 23 19)(15 24 20)(16 21 17)
(1 35 24)(2 36 21)(3 33 22)(4 34 23)(5 12 16)(6 9 13)(7 10 14)(8 11 15)(17 30 26)(18 31 27)(19 32 28)(20 29 25)
(1 8 29)(2 30 5)(3 6 31)(4 32 7)(9 27 33)(10 34 28)(11 25 35)(12 36 26)(13 18 22)(14 23 19)(15 20 24)(16 21 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)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(2 4)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 20)(12 19)(13 27)(14 26)(15 25)(16 28)(21 34)(22 33)(23 36)(24 35)```

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

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

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

C338D4 is a maximal subgroup of
S3×C3⋊D12  C3⋊S34D12  D6.3S32  Dic3.S32  C12.39S32  C12.40S32  C12.73S32  S3×C12⋊S3  C62.93D6  C3⋊S3×C3⋊D4  C6223D6
C338D4 is a maximal quotient of
C338D8  C3316SD16  C3317SD16  C338Q16  C62.78D6  C62.79D6  C62.80D6

33 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4 6A ··· 6E 6F 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 2 2 3 ··· 3 3 3 3 3 4 6 ··· 6 6 6 6 6 6 6 12 ··· 12 size 1 1 18 54 2 ··· 2 4 4 4 4 6 2 ··· 2 4 4 4 4 18 18 6 ··· 6

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D12 C3⋊D4 S32 C3⋊D12 kernel C33⋊8D4 C32×Dic3 C6×C3⋊S3 C2×C33⋊C2 C3×Dic3 C2×C3⋊S3 C33 C3×C6 C32 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 8 2 4 4

Matrix representation of C338D4 in GL8(ℤ)

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

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

C338D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_8D_4`
`% in TeX`

`G:=Group("C3^3:8D4");`
`// GroupNames label`

`G:=SmallGroup(216,129);`
`// by ID`

`G=gap.SmallGroup(216,129);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,201,730,5189]);`
`// Polycyclic`

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

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