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## G = C33⋊8D8order 432 = 24·33

### 5th semidirect product of C33 and D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊8D8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C33⋊8D8
 Lower central C33 — C32×C6 — C32×C12 — C33⋊8D8
 Upper central C1 — C2 — C4

Generators and relations for C338D8
G = < a,b,c,d,e | a3=b3=c3=d8=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: 1792 in 196 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×17], C6, C6 [×4], C6 [×5], C8, D4 [×2], C32, C32 [×4], C32 [×4], C12, C12 [×4], C12 [×4], D6 [×17], C2×C6, D8, C3×S3 [×4], C3⋊S3 [×14], C3×C6, C3×C6 [×4], C3×C6 [×4], C3⋊C8, C24 [×4], D12 [×13], C3×D4, C33, C3×C12, C3×C12 [×4], C3×C12 [×4], S3×C6 [×4], C2×C3⋊S3 [×14], D24 [×4], D4⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C3×C3⋊C8 [×4], C3×C24, C3×D12 [×4], C12⋊S3, C12⋊S3 [×9], C32×C12, C6×C3⋊S3, C2×C33⋊C2, C3⋊D24 [×4], C325D8, C32×C3⋊C8, C3×C12⋊S3, C3312D4, C338D8
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], D8, C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, D24 [×4], D4⋊S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C3⋊D24 [×4], C325D8, C338D4, C338D8

Smallest permutation representation of C338D8
On 72 points
Generators in S72
```(1 38 43)(2 39 44)(3 40 45)(4 33 46)(5 34 47)(6 35 48)(7 36 41)(8 37 42)(9 51 19)(10 52 20)(11 53 21)(12 54 22)(13 55 23)(14 56 24)(15 49 17)(16 50 18)(25 57 68)(26 58 69)(27 59 70)(28 60 71)(29 61 72)(30 62 65)(31 63 66)(32 64 67)
(1 54 57)(2 55 58)(3 56 59)(4 49 60)(5 50 61)(6 51 62)(7 52 63)(8 53 64)(9 30 48)(10 31 41)(11 32 42)(12 25 43)(13 26 44)(14 27 45)(15 28 46)(16 29 47)(17 71 33)(18 72 34)(19 65 35)(20 66 36)(21 67 37)(22 68 38)(23 69 39)(24 70 40)
(1 25 22)(2 23 26)(3 27 24)(4 17 28)(5 29 18)(6 19 30)(7 31 20)(8 21 32)(9 62 35)(10 36 63)(11 64 37)(12 38 57)(13 58 39)(14 40 59)(15 60 33)(16 34 61)(41 66 52)(42 53 67)(43 68 54)(44 55 69)(45 70 56)(46 49 71)(47 72 50)(48 51 65)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 8)(3 7)(4 6)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 72)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)```

`G:=sub<Sym(72)| (1,38,43)(2,39,44)(3,40,45)(4,33,46)(5,34,47)(6,35,48)(7,36,41)(8,37,42)(9,51,19)(10,52,20)(11,53,21)(12,54,22)(13,55,23)(14,56,24)(15,49,17)(16,50,18)(25,57,68)(26,58,69)(27,59,70)(28,60,71)(29,61,72)(30,62,65)(31,63,66)(32,64,67), (1,54,57)(2,55,58)(3,56,59)(4,49,60)(5,50,61)(6,51,62)(7,52,63)(8,53,64)(9,30,48)(10,31,41)(11,32,42)(12,25,43)(13,26,44)(14,27,45)(15,28,46)(16,29,47)(17,71,33)(18,72,34)(19,65,35)(20,66,36)(21,67,37)(22,68,38)(23,69,39)(24,70,40), (1,25,22)(2,23,26)(3,27,24)(4,17,28)(5,29,18)(6,19,30)(7,31,20)(8,21,32)(9,62,35)(10,36,63)(11,64,37)(12,38,57)(13,58,39)(14,40,59)(15,60,33)(16,34,61)(41,66,52)(42,53,67)(43,68,54)(44,55,69)(45,70,56)(46,49,71)(47,72,50)(48,51,65), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,72)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63)>;`

`G:=Group( (1,38,43)(2,39,44)(3,40,45)(4,33,46)(5,34,47)(6,35,48)(7,36,41)(8,37,42)(9,51,19)(10,52,20)(11,53,21)(12,54,22)(13,55,23)(14,56,24)(15,49,17)(16,50,18)(25,57,68)(26,58,69)(27,59,70)(28,60,71)(29,61,72)(30,62,65)(31,63,66)(32,64,67), (1,54,57)(2,55,58)(3,56,59)(4,49,60)(5,50,61)(6,51,62)(7,52,63)(8,53,64)(9,30,48)(10,31,41)(11,32,42)(12,25,43)(13,26,44)(14,27,45)(15,28,46)(16,29,47)(17,71,33)(18,72,34)(19,65,35)(20,66,36)(21,67,37)(22,68,38)(23,69,39)(24,70,40), (1,25,22)(2,23,26)(3,27,24)(4,17,28)(5,29,18)(6,19,30)(7,31,20)(8,21,32)(9,62,35)(10,36,63)(11,64,37)(12,38,57)(13,58,39)(14,40,59)(15,60,33)(16,34,61)(41,66,52)(42,53,67)(43,68,54)(44,55,69)(45,70,56)(46,49,71)(47,72,50)(48,51,65), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,72)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63) );`

`G=PermutationGroup([(1,38,43),(2,39,44),(3,40,45),(4,33,46),(5,34,47),(6,35,48),(7,36,41),(8,37,42),(9,51,19),(10,52,20),(11,53,21),(12,54,22),(13,55,23),(14,56,24),(15,49,17),(16,50,18),(25,57,68),(26,58,69),(27,59,70),(28,60,71),(29,61,72),(30,62,65),(31,63,66),(32,64,67)], [(1,54,57),(2,55,58),(3,56,59),(4,49,60),(5,50,61),(6,51,62),(7,52,63),(8,53,64),(9,30,48),(10,31,41),(11,32,42),(12,25,43),(13,26,44),(14,27,45),(15,28,46),(16,29,47),(17,71,33),(18,72,34),(19,65,35),(20,66,36),(21,67,37),(22,68,38),(23,69,39),(24,70,40)], [(1,25,22),(2,23,26),(3,27,24),(4,17,28),(5,29,18),(6,19,30),(7,31,20),(8,21,32),(9,62,35),(10,36,63),(11,64,37),(12,38,57),(13,58,39),(14,40,59),(15,60,33),(16,34,61),(41,66,52),(42,53,67),(43,68,54),(44,55,69),(45,70,56),(46,49,71),(47,72,50),(48,51,65)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,8),(3,7),(4,6),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,72),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,64),(56,63)])`

60 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4 6A ··· 6E 6F 6G 6H 6I 6J 6K 8A 8B 12A ··· 12H 12I ··· 12Q 24A ··· 24P order 1 2 2 2 3 ··· 3 3 3 3 3 4 6 ··· 6 6 6 6 6 6 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 36 108 2 ··· 2 4 4 4 4 2 2 ··· 2 4 4 4 4 36 36 6 6 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D8 D12 C3⋊D4 D24 S32 D4⋊S3 C3⋊D12 C3⋊D24 kernel C33⋊8D8 C32×C3⋊C8 C3×C12⋊S3 C33⋊12D4 C3×C3⋊C8 C12⋊S3 C32×C6 C3×C12 C33 C3×C6 C3×C6 C32 C12 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 2 8 2 16 4 1 4 8

Matrix representation of C338D8 in GL6(𝔽73)

 1 70 0 0 0 0 1 71 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 71 3 0 0 0 0 72 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
,
 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 72 72 0 0 0 0 1 0
,
 13 15 0 0 0 0 68 28 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 72 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,70,71,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[71,72,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[13,68,0,0,0,0,15,28,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

C338D8 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_8D_8`
`% in TeX`

`G:=Group("C3^3:8D8");`
`// GroupNames label`

`G:=SmallGroup(432,438);`
`// by ID`

`G=gap.SmallGroup(432,438);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,85,135,58,571,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=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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