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## G = He3⋊3Q8order 216 = 23·33

### 1st semidirect product of He3 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — He3⋊3Q8
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C32⋊C12 — He3⋊3Q8
 Lower central C32 — C3×C6 — He3⋊3Q8
 Upper central C1 — C2 — C4

Generators and relations for He33Q8
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, bd=db, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation of He33Q8
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 7 15)(2 8 16)(3 5 13)(4 6 14)(9 27 46)(10 28 47)(11 25 48)(12 26 45)(17 42 65)(18 43 66)(19 44 67)(20 41 68)(21 33 40)(22 34 37)(23 35 38)(24 36 39)(29 49 58)(30 50 59)(31 51 60)(32 52 57)(53 72 62)(54 69 63)(55 70 64)(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)(2,22,30)(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), (1,7,15)(2,8,16)(3,5,13)(4,6,14)(9,27,46)(10,28,47)(11,25,48)(12,26,45)(17,42,65)(18,43,66)(19,44,67)(20,41,68)(21,33,40)(22,34,37)(23,35,38)(24,36,39)(29,49,58)(30,50,59)(31,51,60)(32,52,57)(53,72,62)(54,69,63)(55,70,64)(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)(2,22,30)(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), (1,7,15)(2,8,16)(3,5,13)(4,6,14)(9,27,46)(10,28,47)(11,25,48)(12,26,45)(17,42,65)(18,43,66)(19,44,67)(20,41,68)(21,33,40)(22,34,37)(23,35,38)(24,36,39)(29,49,58)(30,50,59)(31,51,60)(32,52,57)(53,72,62)(54,69,63)(55,70,64)(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),(2,22,30),(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)], [(1,7,15),(2,8,16),(3,5,13),(4,6,14),(9,27,46),(10,28,47),(11,25,48),(12,26,45),(17,42,65),(18,43,66),(19,44,67),(20,41,68),(21,33,40),(22,34,37),(23,35,38),(24,36,39),(29,49,58),(30,50,59),(31,51,60),(32,52,57),(53,72,62),(54,69,63),(55,70,64),(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)]])

He33Q8 is a maximal subgroup of
He33SD16  He32Q16  He34SD16  He33Q16  He34Q16  He36SD16  He38SD16  He36Q16  C3⋊S3⋊Dic6  C12.84S32  C12.85S32  C12.S32  C62.36D6  C62.13D6  Q8×C32⋊C6
He33Q8 is a maximal quotient of
C62.19D6  C62.20D6

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 12A 12B 12C ··· 12J 12K 12L 12M 12N order 1 2 3 3 3 3 3 3 4 4 4 6 6 6 6 6 6 12 12 12 ··· 12 12 12 12 12 size 1 1 2 3 3 6 6 6 2 18 18 2 3 3 6 6 6 2 2 6 ··· 6 18 18 18 18

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + - + - + + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 C3×Dic6 C32⋊C6 C2×C32⋊C6 He3⋊3Q8 kernel He3⋊3Q8 C32⋊C12 C4×He3 C32⋊4Q8 C3⋊Dic3 C3×C12 C3×C12 He3 C3×C6 C12 C32 C32 C6 C3 C4 C2 C1 # reps 1 2 1 2 4 2 1 1 1 2 2 2 2 4 1 1 2

Matrix representation of He33Q8 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
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 0 0 0 12 1 0 0 0 0 12 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0
,
 3 7 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 6 10
,
 0 11 11 2 0 11 11 0 0 2 11 0 11 2 0 11 0 11 0 2 11 0 11 0 0 11 0 11 11 2 11 0 11 0 0 2

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],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,12,12,0,0,0,0,1,0,0,0,0,0],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,11,11,0,0,11,11,0,2,2,11,0,11,0,0,11,0,11,2,2,11,0,11,0,0,11,0,11,11,0,11,0,11,0,2,2] >;

He33Q8 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3Q_8
% in TeX

G:=Group("He3:3Q8");
// GroupNames label

G:=SmallGroup(216,49);
// by ID

G=gap.SmallGroup(216,49);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,1444,736,5189]);
// 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,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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