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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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