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## G = C33⋊7D4order 216 = 23·33

### 4th semidirect product of C33 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊7D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊7D4
 Lower central C33 — C32×C6 — C33⋊7D4
 Upper central C1 — C2

Generators and relations for C337D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 812 in 136 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×14], C6, C6 [×4], C6 [×8], D4, C32, C32 [×4], C32 [×4], Dic3 [×4], C12, D6, D6 [×9], C2×C6 [×4], C3×S3 [×4], C3⋊S3 [×13], C3×C6, C3×C6 [×4], C3×C6 [×5], D12, C3⋊D4 [×4], C33, C3×Dic3 [×4], C3⋊Dic3, S3×C6 [×4], C2×C3⋊S3 [×9], C62, S3×C32, C33⋊C2, C32×C6, C3⋊D12 [×4], C327D4, C3×C3⋊Dic3, S3×C3×C6, C2×C33⋊C2, C337D4
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], C3⋊S3, D12, C3⋊D4 [×4], S32 [×4], C2×C3⋊S3, C3⋊D12 [×4], C327D4, S3×C3⋊S3, C337D4

Smallest permutation representation of C337D4
On 36 points
Generators in S36
```(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 29 21)(2 22 30)(3 31 23)(4 24 32)(5 11 35)(6 36 12)(7 9 33)(8 34 10)(13 28 19)(14 20 25)(15 26 17)(16 18 27)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(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 3)(5 16)(6 15)(7 14)(8 13)(9 25)(10 28)(11 27)(12 26)(17 36)(18 35)(19 34)(20 33)(21 31)(22 30)(23 29)(24 32)```

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

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

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

C337D4 is a maximal subgroup of
S3×C3⋊D12  D64S32  (S3×C6).D6  D6.3S32  D12⋊(C3⋊S3)  C12.73S32  C12.58S32  C3⋊S3×D12  C62.90D6  S3×C327D4  C6223D6
C337D4 is a maximal quotient of
C337D8  C3314SD16  C3315SD16  C337Q16  C62.77D6  C62.79D6  C62.82D6

33 conjugacy classes

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

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⋊7D4 C3×C3⋊Dic3 S3×C3×C6 C2×C33⋊C2 C3⋊Dic3 S3×C6 C33 C3×C6 C32 C32 C6 C3 # reps 1 1 1 1 1 4 1 5 2 8 4 4

Matrix representation of C337D4 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 1 10 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 12 12 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 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 12 12
,
 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
,
 7 3 0 0 0 0 0 0 5 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 1 10 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12
,
 12 0 0 0 0 0 0 0 9 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 12 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[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],[7,5,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[12,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;`

C337D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_7D_4`
`% in TeX`

`G:=Group("C3^3:7D4");`
`// GroupNames label`

`G:=SmallGroup(216,128);`
`// by ID`

`G=gap.SmallGroup(216,128);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,201,730,5189]);`
`// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

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