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## G = C32⋊2D8order 144 = 24·32

### 1st semidirect product of C32 and D8 acting via D8/C4=C22

Aliases: C322D8, D122S3, C12.9D6, C4.8S32, (C3×C6).6D4, C32(D4⋊S3), (C3×D12)⋊1C2, C324C81C2, C6.7(C3⋊D4), (C3×C12).1C22, C2.3(D6⋊S3), SmallGroup(144,56)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊2D8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8
 Lower central C32 — C3×C6 — C3×C12 — C32⋊2D8
 Upper central C1 — C2 — C4

Generators and relations for C322D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Character table of C322D8

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D size 1 1 12 12 2 2 4 2 2 2 4 12 12 12 12 18 18 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 2 2 0 0 2 2 2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 -2 0 2 -1 -1 2 -1 2 -1 0 0 1 1 0 0 2 -1 -1 -1 orthogonal lifted from D6 ρ7 2 2 2 0 2 -1 -1 2 -1 2 -1 0 0 -1 -1 0 0 2 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 2 -1 2 -1 2 2 -1 -1 -1 -1 0 0 0 0 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 2 0 -2 -1 2 -1 2 2 -1 -1 1 1 0 0 0 0 -1 2 -1 -1 orthogonal lifted from D6 ρ10 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ11 2 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 0 2 -1 -1 -2 -1 2 -1 0 0 -√-3 √-3 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ13 2 2 0 0 2 -1 -1 -2 -1 2 -1 0 0 √-3 -√-3 0 0 -2 1 1 1 complex lifted from C3⋊D4 ρ14 2 2 0 0 -1 2 -1 -2 2 -1 -1 √-3 -√-3 0 0 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 0 -1 2 -1 -2 2 -1 -1 -√-3 √-3 0 0 0 0 1 -2 1 1 complex lifted from C3⋊D4 ρ16 4 -4 0 0 4 -2 -2 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 0 0 -2 4 -2 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ18 4 4 0 0 -2 -2 1 4 -2 -2 1 0 0 0 0 0 0 -2 -2 1 1 orthogonal lifted from S32 ρ19 4 4 0 0 -2 -2 1 -4 -2 -2 1 0 0 0 0 0 0 2 2 -1 -1 symplectic lifted from D6⋊S3, Schur index 2 ρ20 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 0 0 0 0 3i -3i complex faithful ρ21 4 -4 0 0 -2 -2 1 0 2 2 -1 0 0 0 0 0 0 0 0 -3i 3i complex faithful

Smallest permutation representation of C322D8
On 48 points
Generators in S48
```(1 35 46)(2 47 36)(3 37 48)(4 41 38)(5 39 42)(6 43 40)(7 33 44)(8 45 34)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(1 46 35)(2 36 47)(3 48 37)(4 38 41)(5 42 39)(6 40 43)(7 44 33)(8 34 45)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)```

`G:=sub<Sym(48)| (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)>;`

`G:=Group( (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43) );`

`G=PermutationGroup([(1,35,46),(2,47,36),(3,37,48),(4,41,38),(5,39,42),(6,43,40),(7,33,44),(8,45,34),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(1,46,35),(2,36,47),(3,48,37),(4,38,41),(5,42,39),(6,40,43),(7,44,33),(8,34,45),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)])`

C322D8 is a maximal subgroup of
C32⋊D16  C32⋊SD32  C244D6  C246D6  D12.2D6  D12.30D6  D1220D6  S3×D4⋊S3  D129D6  D126D6  D12.12D6  D36⋊S3  He33D8  C336D8  C339D8
C322D8 is a maximal quotient of
C322D16  D24.S3  C322Q32  D123Dic3  C12.8Dic6  D36⋊S3  He32D8  C336D8  C339D8

Matrix representation of C322D8 in GL4(𝔽5) generated by

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

C322D8 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_2D_8`
`% in TeX`

`G:=Group("C3^2:2D8");`
`// GroupNames label`

`G:=SmallGroup(144,56);`
`// by ID`

`G=gap.SmallGroup(144,56);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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