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## G = C32⋊4Q8order 72 = 23·32

### 2nd semidirect product of C32 and Q8 acting via Q8/C4=C2

Aliases: C324Q8, C12.3S3, C32Dic6, C6.12D6, C4.(C3⋊S3), (C3×C12).1C2, C3⋊Dic3.3C2, (C3×C6).11C22, C2.3(C2×C3⋊S3), SmallGroup(72,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3⋊Dic3 — C32⋊4Q8
 Lower central C32 — C3×C6 — C32⋊4Q8
 Upper central C1 — C2 — C4

Generators and relations for C324Q8
G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Character table of C324Q8

 class 1 2 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 2 2 2 2 18 18 2 2 2 2 2 2 2 2 2 2 2 2 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 -1 -1 2 -1 2 0 0 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -1 -1 -1 2 2 0 0 -1 2 -1 -1 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ7 2 2 -1 2 -1 -1 -2 0 0 -1 -1 2 -1 1 -2 1 1 1 -2 1 1 orthogonal lifted from D6 ρ8 2 2 -1 2 -1 -1 2 0 0 -1 -1 2 -1 -1 2 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 2 2 -1 -1 -1 -2 0 0 2 -1 -1 -1 -2 1 1 -2 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 -1 -1 2 -1 -2 0 0 -1 -1 -1 2 1 1 -2 1 -2 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 -1 -1 -1 2 0 0 2 -1 -1 -1 2 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -1 -1 -1 2 -2 0 0 -1 2 -1 -1 1 1 1 1 1 1 -2 -2 orthogonal lifted from D6 ρ13 2 -2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -1 -1 -1 2 0 0 0 1 -2 1 1 -√3 -√3 -√3 √3 √3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ15 2 -2 -1 2 -1 -1 0 0 0 1 1 -2 1 -√3 0 √3 √3 -√3 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ16 2 -2 -1 -1 2 -1 0 0 0 1 1 1 -2 -√3 √3 0 √3 0 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 2 -1 -1 -1 0 0 0 -2 1 1 1 0 -√3 √3 0 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 2 -1 -1 -1 0 0 0 -2 1 1 1 0 √3 -√3 0 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ19 2 -2 -1 -1 -1 2 0 0 0 1 -2 1 1 √3 √3 √3 -√3 -√3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 -1 2 -1 -1 0 0 0 1 1 -2 1 √3 0 -√3 -√3 √3 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 -1 -1 2 -1 0 0 0 1 1 1 -2 √3 -√3 0 -√3 0 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2

Smallest permutation representation of C324Q8
Regular action on 72 points
Generators in S72
```(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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)
(1 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)```

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

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

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

C324Q8 is a maximal subgroup of
D12.S3  C323Q16  C242S3  C325Q16  C329SD16  C327Q16  S3×Dic6  D125S3  C12.59D6  C12.D6  Q8×C3⋊S3  He33Q8  C12.D9  C334Q8  C338Q8  A4⋊Dic6  C12.6S4  C15⋊Dic6  C12.D15
C324Q8 is a maximal quotient of
C6.Dic6  C12⋊Dic3  C12.D9  He34Q8  C334Q8  C338Q8  A4⋊Dic6  C15⋊Dic6  C12.D15

Matrix representation of C324Q8 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 5 3
,
 0 1 0 0 12 12 0 0 0 0 3 0 0 0 8 9
,
 3 6 0 0 7 10 0 0 0 0 12 0 0 0 0 12
,
 5 0 0 0 8 8 0 0 0 0 2 8 0 0 11 11
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,5,0,0,0,3],[0,12,0,0,1,12,0,0,0,0,3,8,0,0,0,9],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,0,8,0,0,0,0,2,11,0,0,8,11] >;`

C324Q8 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_4Q_8`
`% in TeX`

`G:=Group("C3^2:4Q8");`
`// GroupNames label`

`G:=SmallGroup(72,31);`
`// by ID`

`G=gap.SmallGroup(72,31);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,323,1204]);`
`// Polycyclic`

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

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