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## G = He3⋊2Q8order 216 = 23·33

### The semidirect product of He3 and Q8 acting via Q8/C2=C22

Aliases: He32Q8, C32⋊Dic6, C6.16S32, (C3×C6).1D6, C3⋊Dic3.S3, C32⋊C12.1C2, He33C4.2C2, C2.3(C32⋊D6), C3.2(C322Q8), (C2×He3).1C22, SmallGroup(216,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊2Q8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C32⋊C12 — He3⋊2Q8
 Lower central He3 — C2×He3 — He3⋊2Q8
 Upper central C1 — C2

Generators and relations for He32Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Character table of He32Q8

 class 1 2 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F size 1 1 2 6 6 12 18 18 18 2 6 6 12 18 18 18 18 18 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 -1 2 -1 0 0 -2 2 2 -1 -1 0 1 1 0 0 0 orthogonal lifted from D6 ρ6 2 2 2 -1 2 -1 0 0 2 2 2 -1 -1 0 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 2 2 -1 -1 0 -2 0 2 -1 2 -1 1 0 0 1 0 0 orthogonal lifted from D6 ρ8 2 2 2 2 -1 -1 0 2 0 2 -1 2 -1 -1 0 0 -1 0 0 orthogonal lifted from S3 ρ9 2 -2 2 2 2 2 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ10 2 -2 2 -1 2 -1 0 0 0 -2 -2 1 1 0 √3 -√3 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ11 2 -2 2 -1 2 -1 0 0 0 -2 -2 1 1 0 -√3 √3 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ12 2 -2 2 2 -1 -1 0 0 0 -2 1 -2 1 -√3 0 0 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ13 2 -2 2 2 -1 -1 0 0 0 -2 1 -2 1 √3 0 0 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ14 4 4 4 -2 -2 1 0 0 0 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ15 4 -4 4 -2 -2 1 0 0 0 -4 2 2 -1 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2 ρ16 6 6 -3 0 0 0 -2 0 0 -3 0 0 0 0 0 0 0 1 1 orthogonal lifted from C32⋊D6 ρ17 6 6 -3 0 0 0 2 0 0 -3 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from C32⋊D6 ρ18 6 -6 -3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 -√3 √3 symplectic faithful, Schur index 2 ρ19 6 -6 -3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 √3 -√3 symplectic faithful, Schur index 2

Smallest permutation representation of He32Q8
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 30 22)(3 23 31)(4 32 24)(5 35 51)(6 52 36)(7 33 49)(8 50 34)(9 69 43)(10 44 70)(11 71 41)(12 42 72)(13 38 60)(14 57 39)(15 40 58)(16 59 37)(17 53 45)(18 46 54)(19 55 47)(20 48 56)(25 61 68)(26 65 62)(27 63 66)(28 67 64)
(1 7 15)(2 16 8)(3 5 13)(4 14 6)(9 27 46)(10 47 28)(11 25 48)(12 45 26)(17 65 42)(18 43 66)(19 67 44)(20 41 68)(21 33 40)(22 37 34)(23 35 38)(24 39 36)(29 49 58)(30 59 50)(31 51 60)(32 57 52)(53 62 72)(54 69 63)(55 64 70)(56 71 61)
(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)(2,30,22)(3,23,31)(4,32,24)(5,35,51)(6,52,36)(7,33,49)(8,50,34)(9,69,43)(10,44,70)(11,71,41)(12,42,72)(13,38,60)(14,57,39)(15,40,58)(16,59,37)(17,53,45)(18,46,54)(19,55,47)(20,48,56)(25,61,68)(26,65,62)(27,63,66)(28,67,64), (1,7,15)(2,16,8)(3,5,13)(4,14,6)(9,27,46)(10,47,28)(11,25,48)(12,45,26)(17,65,42)(18,43,66)(19,67,44)(20,41,68)(21,33,40)(22,37,34)(23,35,38)(24,39,36)(29,49,58)(30,59,50)(31,51,60)(32,57,52)(53,62,72)(54,69,63)(55,64,70)(56,71,61), (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)(2,30,22)(3,23,31)(4,32,24)(5,35,51)(6,52,36)(7,33,49)(8,50,34)(9,69,43)(10,44,70)(11,71,41)(12,42,72)(13,38,60)(14,57,39)(15,40,58)(16,59,37)(17,53,45)(18,46,54)(19,55,47)(20,48,56)(25,61,68)(26,65,62)(27,63,66)(28,67,64), (1,7,15)(2,16,8)(3,5,13)(4,14,6)(9,27,46)(10,47,28)(11,25,48)(12,45,26)(17,65,42)(18,43,66)(19,67,44)(20,41,68)(21,33,40)(22,37,34)(23,35,38)(24,39,36)(29,49,58)(30,59,50)(31,51,60)(32,57,52)(53,62,72)(54,69,63)(55,64,70)(56,71,61), (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),(2,30,22),(3,23,31),(4,32,24),(5,35,51),(6,52,36),(7,33,49),(8,50,34),(9,69,43),(10,44,70),(11,71,41),(12,42,72),(13,38,60),(14,57,39),(15,40,58),(16,59,37),(17,53,45),(18,46,54),(19,55,47),(20,48,56),(25,61,68),(26,65,62),(27,63,66),(28,67,64)], [(1,7,15),(2,16,8),(3,5,13),(4,14,6),(9,27,46),(10,47,28),(11,25,48),(12,45,26),(17,65,42),(18,43,66),(19,67,44),(20,41,68),(21,33,40),(22,37,34),(23,35,38),(24,39,36),(29,49,58),(30,59,50),(31,51,60),(32,57,52),(53,62,72),(54,69,63),(55,64,70),(56,71,61)], [(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)]])

He32Q8 is a maximal subgroup of   He32SD16  He3⋊Q16  C3⋊S3⋊Dic6  C12.91S32  C12.85S32  C62.8D6  C62.9D6
He32Q8 is a maximal quotient of   C62.D6  C62.3D6

Matrix representation of He32Q8 in GL6(𝔽13)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 0 0 0 0 0 1 0 0 0 0 12 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0
,
 2 0 0 2 0 2 11 11 2 0 2 0 0 2 2 0 0 2 2 0 11 11 2 0 0 2 0 2 2 0 2 0 2 0 11 11
,
 12 6 7 1 12 6 7 1 7 6 7 1 7 1 12 6 12 6 7 6 7 1 7 1 12 6 12 6 7 1 7 1 7 1 7 6

G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,12,0,0,0,0,1,12,0,0,0,0],[2,11,0,2,0,2,0,11,2,0,2,0,0,2,2,11,0,2,2,0,0,11,2,0,0,2,0,2,2,11,2,0,2,0,0,11],[12,7,7,7,12,7,6,1,1,6,6,1,7,7,12,7,12,7,1,6,6,1,6,1,12,7,12,7,7,7,6,1,6,1,1,6] >;

He32Q8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2Q_8
% in TeX

G:=Group("He3:2Q8");
// GroupNames label

G:=SmallGroup(216,33);
// by ID

G=gap.SmallGroup(216,33);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,201,1444,382,5189,2603]);
// Polycyclic

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

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