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## G = He3⋊3D9order 486 = 2·35

### 1st semidirect product of He3 and D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — He3⋊3D9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9×He3 — He3⋊3D9
 Lower central C32×C9 — He3⋊3D9
 Upper central C1

Generators and relations for He33D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1124 in 118 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, He3, He3, C33, C33, C3×D9, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C3×He3, C32⋊D9, C3×C9⋊S3, He34S3, C324D9, C9×He3, He33D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C32⋊C6, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, He34S3, He3.4S3, He33D9

Smallest permutation representation of He33D9
On 81 points
Generators in S81
(1 16 68)(2 17 69)(3 18 70)(4 10 71)(5 11 72)(6 12 64)(7 13 65)(8 14 66)(9 15 67)(19 37 49)(20 38 50)(21 39 51)(22 40 52)(23 41 53)(24 42 54)(25 43 46)(26 44 47)(27 45 48)(28 79 63)(29 80 55)(30 81 56)(31 73 57)(32 74 58)(33 75 59)(34 76 60)(35 77 61)(36 78 62)
(1 77 53)(2 78 54)(3 79 46)(4 80 47)(5 81 48)(6 73 49)(7 74 50)(8 75 51)(9 76 52)(10 55 26)(11 56 27)(12 57 19)(13 58 20)(14 59 21)(15 60 22)(16 61 23)(17 62 24)(18 63 25)(28 43 70)(29 44 71)(30 45 72)(31 37 64)(32 38 65)(33 39 66)(34 40 67)(35 41 68)(36 42 69)
(10 55 26)(11 56 27)(12 57 19)(13 58 20)(14 59 21)(15 60 22)(16 61 23)(17 62 24)(18 63 25)(28 70 43)(29 71 44)(30 72 45)(31 64 37)(32 65 38)(33 66 39)(34 67 40)(35 68 41)(36 69 42)
(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)(73 74 75 76 77 78 79 80 81)
(1 9)(2 8)(3 7)(4 6)(10 64)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 29)(20 28)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(37 55)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 74)(47 73)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)

G:=sub<Sym(81)| (1,16,68)(2,17,69)(3,18,70)(4,10,71)(5,11,72)(6,12,64)(7,13,65)(8,14,66)(9,15,67)(19,37,49)(20,38,50)(21,39,51)(22,40,52)(23,41,53)(24,42,54)(25,43,46)(26,44,47)(27,45,48)(28,79,63)(29,80,55)(30,81,56)(31,73,57)(32,74,58)(33,75,59)(34,76,60)(35,77,61)(36,78,62), (1,77,53)(2,78,54)(3,79,46)(4,80,47)(5,81,48)(6,73,49)(7,74,50)(8,75,51)(9,76,52)(10,55,26)(11,56,27)(12,57,19)(13,58,20)(14,59,21)(15,60,22)(16,61,23)(17,62,24)(18,63,25)(28,43,70)(29,44,71)(30,45,72)(31,37,64)(32,38,65)(33,39,66)(34,40,67)(35,41,68)(36,42,69), (10,55,26)(11,56,27)(12,57,19)(13,58,20)(14,59,21)(15,60,22)(16,61,23)(17,62,24)(18,63,25)(28,70,43)(29,71,44)(30,72,45)(31,64,37)(32,65,38)(33,66,39)(34,67,40)(35,68,41)(36,69,42), (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)(73,74,75,76,77,78,79,80,81), (1,9)(2,8)(3,7)(4,6)(10,64)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,29)(20,28)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(37,55)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,74)(47,73)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)>;

G:=Group( (1,16,68)(2,17,69)(3,18,70)(4,10,71)(5,11,72)(6,12,64)(7,13,65)(8,14,66)(9,15,67)(19,37,49)(20,38,50)(21,39,51)(22,40,52)(23,41,53)(24,42,54)(25,43,46)(26,44,47)(27,45,48)(28,79,63)(29,80,55)(30,81,56)(31,73,57)(32,74,58)(33,75,59)(34,76,60)(35,77,61)(36,78,62), (1,77,53)(2,78,54)(3,79,46)(4,80,47)(5,81,48)(6,73,49)(7,74,50)(8,75,51)(9,76,52)(10,55,26)(11,56,27)(12,57,19)(13,58,20)(14,59,21)(15,60,22)(16,61,23)(17,62,24)(18,63,25)(28,43,70)(29,44,71)(30,45,72)(31,37,64)(32,38,65)(33,39,66)(34,40,67)(35,41,68)(36,42,69), (10,55,26)(11,56,27)(12,57,19)(13,58,20)(14,59,21)(15,60,22)(16,61,23)(17,62,24)(18,63,25)(28,70,43)(29,71,44)(30,72,45)(31,64,37)(32,65,38)(33,66,39)(34,67,40)(35,68,41)(36,69,42), (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)(73,74,75,76,77,78,79,80,81), (1,9)(2,8)(3,7)(4,6)(10,64)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,29)(20,28)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(37,55)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,74)(47,73)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75) );

G=PermutationGroup([[(1,16,68),(2,17,69),(3,18,70),(4,10,71),(5,11,72),(6,12,64),(7,13,65),(8,14,66),(9,15,67),(19,37,49),(20,38,50),(21,39,51),(22,40,52),(23,41,53),(24,42,54),(25,43,46),(26,44,47),(27,45,48),(28,79,63),(29,80,55),(30,81,56),(31,73,57),(32,74,58),(33,75,59),(34,76,60),(35,77,61),(36,78,62)], [(1,77,53),(2,78,54),(3,79,46),(4,80,47),(5,81,48),(6,73,49),(7,74,50),(8,75,51),(9,76,52),(10,55,26),(11,56,27),(12,57,19),(13,58,20),(14,59,21),(15,60,22),(16,61,23),(17,62,24),(18,63,25),(28,43,70),(29,44,71),(30,45,72),(31,37,64),(32,38,65),(33,39,66),(34,40,67),(35,41,68),(36,42,69)], [(10,55,26),(11,56,27),(12,57,19),(13,58,20),(14,59,21),(15,60,22),(16,61,23),(17,62,24),(18,63,25),(28,70,43),(29,71,44),(30,72,45),(31,64,37),(32,65,38),(33,66,39),(34,67,40),(35,68,41),(36,69,42)], [(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),(73,74,75,76,77,78,79,80,81)], [(1,9),(2,8),(3,7),(4,6),(10,64),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,29),(20,28),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(37,55),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,74),(47,73),(48,81),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75)]])

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G ··· 3Q 6A 6B 9A ··· 9I 9J ··· 9AG order 1 2 3 3 3 3 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 size 1 81 2 2 2 2 3 3 6 ··· 6 81 81 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 6 6 type + + + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 D9 C3×S3 C3×D9 C32⋊C6 He3.4S3 kernel He3⋊3D9 C9×He3 C32⋊4D9 C32×C9 C32⋊C9 C32×C9 C3×He3 C3×C9 He3 C33 C32 C9 C3 # reps 1 1 2 2 2 1 1 6 9 2 18 3 6

Matrix representation of He33D9 in GL8(𝔽19)

 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 18 18 18 0 0 18 1 1 0 1 17 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 1 18
,
 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 1 1 0 0 18 0 0 18 1 0 1 1 18
,
 7 14 0 0 0 0 0 0 5 2 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 5 2 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 5 2 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 5 2
,
 14 17 0 0 0 0 0 0 12 5 0 0 0 0 0 0 0 0 7 14 0 0 0 0 0 0 2 12 0 0 0 0 0 0 5 2 2 12 7 3 0 0 7 14 14 17 10 12 0 0 0 0 0 0 2 12 0 0 0 0 0 0 14 17

G:=sub<GL(8,GF(19))| [18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,18,0,0,0,0,0,0,0,18,1,18,18,0,0,0,0,18,17,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,18,0,0,0,1,0,0,1,1,0,0,0,0,18,18,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18],[7,5,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,14,2],[14,12,0,0,0,0,0,0,17,5,0,0,0,0,0,0,0,0,7,2,5,7,0,0,0,0,14,12,2,14,0,0,0,0,0,0,2,14,0,0,0,0,0,0,12,17,0,0,0,0,0,0,7,10,2,14,0,0,0,0,3,12,12,17] >;

He33D9 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3D_9
% in TeX

G:=Group("He3:3D9");
// GroupNames label

G:=SmallGroup(486,142);
// by ID

G=gap.SmallGroup(486,142);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,986,867,873,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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