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## G = C32⋊3Q16order 144 = 24·32

### 2nd semidirect product of C32 and Q16 acting via Q16/C4=C22

Aliases: C323Q16, C32Dic12, C12.15D6, C6.15D12, Dic6.1S3, C3⋊C8.S3, C4.4S32, (C3×C6).12D4, C31(C3⋊Q16), C6.4(C3⋊D4), (C3×C12).7C22, (C3×Dic6).2C2, C2.7(C3⋊D12), C324Q8.2C2, (C3×C3⋊C8).1C2, SmallGroup(144,62)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊3Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊3Q16
 Lower central C32 — C3×C6 — C3×C12 — C32⋊3Q16
 Upper central C1 — C2 — C4

Generators and relations for C323Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Character table of C323Q16

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D size 1 1 2 2 4 2 12 36 2 2 4 6 6 2 2 4 4 4 12 12 6 6 6 6 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 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 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 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 linear of order 2 ρ5 2 2 2 -1 -1 2 0 0 -1 2 -1 -2 -2 -1 -1 2 -1 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ6 2 2 -1 2 -1 2 2 0 2 -1 -1 0 0 2 2 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 2 -1 2 -1 2 -2 0 2 -1 -1 0 0 2 2 -1 -1 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ8 2 2 2 2 2 -2 0 0 2 2 2 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ9 2 2 2 -1 -1 2 0 0 -1 2 -1 2 2 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 -1 -1 -2 0 0 -1 2 -1 0 0 1 1 -2 1 1 0 0 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ11 2 2 2 -1 -1 -2 0 0 -1 2 -1 0 0 1 1 -2 1 1 0 0 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ12 2 -2 2 2 2 0 0 0 -2 -2 -2 -√2 √2 0 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 2 2 0 0 0 -2 -2 -2 √2 -√2 0 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 2 -1 -1 0 0 0 1 -2 1 -√2 √2 -√3 √3 0 -√3 √3 0 0 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 symplectic lifted from Dic12, Schur index 2 ρ15 2 -2 2 -1 -1 0 0 0 1 -2 1 √2 -√2 √3 -√3 0 √3 -√3 0 0 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 symplectic lifted from Dic12, Schur index 2 ρ16 2 -2 2 -1 -1 0 0 0 1 -2 1 √2 -√2 -√3 √3 0 -√3 √3 0 0 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 symplectic lifted from Dic12, Schur index 2 ρ17 2 -2 2 -1 -1 0 0 0 1 -2 1 -√2 √2 √3 -√3 0 √3 -√3 0 0 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 symplectic lifted from Dic12, Schur index 2 ρ18 2 2 -1 2 -1 -2 0 0 2 -1 -1 0 0 -2 -2 1 1 1 √-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ19 2 2 -1 2 -1 -2 0 0 2 -1 -1 0 0 -2 -2 1 1 1 -√-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ20 4 4 -2 -2 1 4 0 0 -2 -2 1 0 0 -2 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ21 4 4 -2 -2 1 -4 0 0 -2 -2 1 0 0 2 2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ22 4 -4 -2 4 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ23 4 -4 -2 -2 1 0 0 0 2 2 -1 0 0 2√3 -2√3 0 -√3 √3 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 4 -4 -2 -2 1 0 0 0 2 2 -1 0 0 -2√3 2√3 0 √3 -√3 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C323Q16
On 48 points
Generators in S48
```(1 35 25)(2 26 36)(3 37 27)(4 28 38)(5 39 29)(6 30 40)(7 33 31)(8 32 34)(9 17 43)(10 44 18)(11 19 45)(12 46 20)(13 21 47)(14 48 22)(15 23 41)(16 42 24)
(1 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 43 17)(10 44 18)(11 45 19)(12 46 20)(13 47 21)(14 48 22)(15 41 23)(16 42 24)
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)```

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

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

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

C323Q16 is a maximal subgroup of
S3×Dic12  C24.3D6  Dic12⋊S3  D6.1D12  D12.27D6  D12.29D6  Dic6.29D6  Dic6.19D6  Dic6.D6  D12.22D6  D12.8D6  S3×C3⋊Q16  Dic6.9D6  D12.24D6  D12.15D6  C3⋊Dic36  C9⋊Dic12  He32Q16  He33Q16  C337Q16  C338Q16  C339Q16
C323Q16 is a maximal quotient of
C6.Dic12  C12.73D12  C6.18D24  C3⋊Dic36  C9⋊Dic12  He33Q16  C337Q16  C338Q16  C339Q16

Matrix representation of C323Q16 in GL6(𝔽73)

 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 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 41 0 0 0 0 16 41 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 19 13 0 0 0 0 62 54 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[19,62,0,0,0,0,13,54,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C323Q16 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_3Q_{16}`
`% in TeX`

`G:=Group("C3^2:3Q16");`
`// GroupNames label`

`G:=SmallGroup(144,62);`
`// by ID`

`G=gap.SmallGroup(144,62);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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