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G = He33D9order 486 = 2·35

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

metabelian, supersoluble, monomial

Aliases: He33D9, (C9×He3)⋊4C2, C321(C3×D9), C32⋊C914S3, C93(C32⋊C6), C321(C9⋊S3), (C32×C9)⋊13S3, (C32×C9)⋊11C6, C324D93C3, (C3×He3).22S3, C33.61(C3×S3), C33.27(C3⋊S3), C3.2(He34S3), C3.2(He3.4S3), C3.4(C3×C9⋊S3), (C3×C9).58(C3×S3), C32.33(C3×C3⋊S3), SmallGroup(486,142)

Series: Derived Chief Lower central Upper central

C1C32×C9 — He33D9
C1C3C32C33C32×C9C9×He3 — He33D9
C32×C9 — He33D9
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 3A3B3C3D3E3F3G···3Q6A6B9A···9I9J···9AG
order123333333···3669···99···9
size1812222336···681812···26···6

54 irreducible representations

dim1111222222266
type++++++++
imageC1C2C3C6S3S3S3C3×S3D9C3×S3C3×D9C32⋊C6He3.4S3
kernelHe33D9C9×He3C324D9C32×C9C32⋊C9C32×C9C3×He3C3×C9He3C33C32C9C3
# reps11222116921836

Matrix representation of He33D9 in GL8(𝔽19)

181000000
180000000
00001000
00000100
00011181818
0018110117
000000181
000000180
,
10000000
01000000
000180000
001180000
000001800
000011800
000000018
000000118
,
110000000
011000000
00100000
00010000
000018100
000018000
000110018
0018101118
,
714000000
52000000
007140000
00520000
000071400
00005200
000000714
00000052
,
1417000000
125000000
007140000
002120000
005221273
0071414171012
000000212
0000001417

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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