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## G = C33⋊D8order 432 = 24·33

### 2nd semidirect product of C33 and D8 acting via D8/C2=D4

Aliases: C332D8, C6.15S3≀C2, C32(C32⋊D8), D6⋊S31S3, C334C81C2, C339D46C2, (C32×C6).9D4, C323(D4⋊S3), C3⋊Dic3.10D6, C2.4(C33⋊D4), (C3×D6⋊S3)⋊1C2, (C3×C6).15(C3⋊D4), (C3×C3⋊Dic3).7C22, SmallGroup(432,582)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊Dic3 — C33⋊D8
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C33⋊D8
 Lower central C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊D8
 Upper central C1 — C2

Generators and relations for C33⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=b, bc=cb, dbd-1=ebe=a, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 684 in 96 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×5], C6, C6 [×9], C8, D4 [×2], C32, C32 [×4], Dic3 [×2], C12, D6 [×4], C2×C6 [×4], D8, C3×S3 [×8], C3⋊S3, C3×C6, C3×C6 [×5], C3⋊C8, D12, C3⋊D4 [×2], C3×D4, C33, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×6], C2×C3⋊S3, C62, D4⋊S3, S3×C32, C3×C3⋊S3, C32×C6, C322C8, D6⋊S3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C32⋊D8, C334C8, C3×D6⋊S3, C339D4, C33⋊D8
Quotients: C1, C2 [×3], C22, S3, D4, D6, D8, C3⋊D4, D4⋊S3, S3≀C2, C32⋊D8, C33⋊D4, C33⋊D8

Character table of C33⋊D8

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 8A 8B 12 size 1 1 12 36 2 4 4 4 4 8 18 2 4 4 4 4 8 12 12 12 12 12 12 12 12 36 36 54 54 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 2 2 2 0 -1 -1 2 2 -1 -1 2 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ6 2 2 0 0 2 2 2 2 2 2 -2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ7 2 2 -2 0 -1 -1 2 2 -1 -1 2 -1 2 -1 -1 2 -1 1 -2 1 1 -2 1 1 1 0 0 0 0 -1 orthogonal lifted from D6 ρ8 2 -2 0 0 2 2 2 2 2 2 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 0 orthogonal lifted from D8 ρ9 2 -2 0 0 2 2 2 2 2 2 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 0 orthogonal lifted from D8 ρ10 2 2 0 0 -1 -1 2 2 -1 -1 -2 -1 2 -1 -1 2 -1 -√-3 0 √-3 -√-3 0 √-3 -√-3 √-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ11 2 2 0 0 -1 -1 2 2 -1 -1 -2 -1 2 -1 -1 2 -1 √-3 0 -√-3 √-3 0 -√-3 √-3 -√-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ12 4 -4 0 0 -2 -2 4 4 -2 -2 0 2 -4 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ13 4 4 0 -2 4 -2 1 -2 -2 1 0 4 1 -2 -2 -2 1 0 0 0 0 0 0 0 0 1 1 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 0 2 4 -2 1 -2 -2 1 0 4 1 -2 -2 -2 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 -2 0 4 1 -2 1 1 -2 0 4 -2 1 1 1 -2 -2 1 1 1 1 1 1 -2 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 2 0 4 1 -2 1 1 -2 0 4 -2 1 1 1 -2 2 -1 -1 -1 -1 -1 -1 2 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 -4 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 2 2 1+3√-3/2 1-3√-3/2 -1 -1 0 -√-3 -3+√-3/2 3+√-3/2 √-3 3-√-3/2 -3-√-3/2 0 0 0 0 0 0 complex faithful ρ18 4 -4 0 0 4 1 -2 1 1 -2 0 -4 2 -1 -1 -1 2 0 -√-3 -√-3 -√-3 √-3 √-3 √-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ19 4 4 -2 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 -2 -2 -1+3√-3/2 -1-3√-3/2 1 1 1+√-3 1 ζ3 ζ32 1 ζ3 ζ32 1-√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ20 4 4 2 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 -2 -2 -1+3√-3/2 -1-3√-3/2 1 1 -1-√-3 -1 ζ65 ζ6 -1 ζ65 ζ6 -1+√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ21 4 -4 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 2 2 1-3√-3/2 1+3√-3/2 -1 -1 0 √-3 -3-√-3/2 3-√-3/2 -√-3 3+√-3/2 -3+√-3/2 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 2 2 1-3√-3/2 1+3√-3/2 -1 -1 0 -√-3 3+√-3/2 -3+√-3/2 √-3 -3-√-3/2 3-√-3/2 0 0 0 0 0 0 complex faithful ρ23 4 4 2 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 -2 -2 -1-3√-3/2 -1+3√-3/2 1 1 -1+√-3 -1 ζ6 ζ65 -1 ζ6 ζ65 -1-√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ24 4 -4 0 0 4 -2 1 -2 -2 1 0 -4 -1 2 2 2 -1 0 0 0 0 0 0 0 0 -√-3 √-3 0 0 0 complex lifted from C32⋊D8 ρ25 4 -4 0 0 4 -2 1 -2 -2 1 0 -4 -1 2 2 2 -1 0 0 0 0 0 0 0 0 √-3 -√-3 0 0 0 complex lifted from C32⋊D8 ρ26 4 -4 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 2 2 1+3√-3/2 1-3√-3/2 -1 -1 0 √-3 3-√-3/2 -3-√-3/2 -√-3 -3+√-3/2 3+√-3/2 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 4 1 -2 1 1 -2 0 -4 2 -1 -1 -1 2 0 √-3 √-3 √-3 -√-3 -√-3 -√-3 0 0 0 0 0 0 complex lifted from C32⋊D8 ρ28 4 4 -2 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 -2 -2 -1-3√-3/2 -1+3√-3/2 1 1 1-√-3 1 ζ32 ζ3 1 ζ32 ζ3 1+√-3 0 0 0 0 0 complex lifted from C33⋊D4 ρ29 8 8 0 0 -4 2 2 -4 2 -1 0 -4 2 2 2 -4 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊D4 ρ30 8 -8 0 0 -4 2 2 -4 2 -1 0 4 -2 -2 -2 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C33⋊D8
On 24 points - transitive group 24T1290
Generators in S24
```(2 22 11)(4 13 24)(6 18 15)(8 9 20)
(1 10 21)(3 23 12)(5 14 17)(7 19 16)
(1 21 10)(2 11 22)(3 23 12)(4 13 24)(5 17 14)(6 15 18)(7 19 16)(8 9 20)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)```

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

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

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

`G:=TransitiveGroup(24,1290);`

On 24 points - transitive group 24T1314
Generators in S24
```(1 24 10)(2 17 11)(3 12 18)(4 13 19)(5 20 14)(6 21 15)(7 16 22)(8 9 23)
(1 10 24)(2 17 11)(3 18 12)(4 13 19)(5 14 20)(6 21 15)(7 22 16)(8 9 23)
(1 24 10)(2 11 17)(3 18 12)(4 13 19)(5 20 14)(6 15 21)(7 22 16)(8 9 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)```

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

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

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

`G:=TransitiveGroup(24,1314);`

Matrix representation of C33⋊D8 in GL4(𝔽7) generated by

 1 0 4 0 5 6 1 4 4 4 0 6 0 0 0 1
,
 6 2 1 1 2 6 6 1 0 0 1 0 0 0 0 2
,
 3 1 4 5 1 3 3 5 0 0 4 0 0 0 0 2
,
 6 5 0 1 0 0 3 4 1 6 3 2 1 1 6 5
,
 6 1 1 1 0 3 4 4 1 6 3 2 6 6 1 2
`G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,1,1,5,0,6,1,0,3,3,6,1,4,2,5],[6,0,1,6,1,3,6,6,1,4,3,1,1,4,2,2] >;`

C33⋊D8 in GAP, Magma, Sage, TeX

`C_3^3\rtimes D_8`
`% in TeX`

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

`G:=SmallGroup(432,582);`
`// by ID`

`G=gap.SmallGroup(432,582);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1684,571,298,677,1027,14118]);`
`// 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,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

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