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## G = He3⋊2D9order 486 = 2·35

### 2nd semidirect product of He3 and D9 acting via D9/C3=S3

Aliases: He32D9, He3⋊C93C2, C32⋊C98S3, (C32×C9)⋊11S3, (C3×He3).6S3, C32.1(C9⋊S3), C33.23(C3⋊S3), C3.7(C33⋊S3), C3.5(He3⋊S3), C3.7(C322D9), C3.7(He3.3S3), C32.23(He3⋊C2), SmallGroup(486,56)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3⋊C9 — He3⋊2D9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — He3⋊C9 — He3⋊2D9
 Lower central He3⋊C9 — He3⋊2D9
 Upper central C1

Generators and relations for He32D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, cac-1=eae=ab-1, ad=da, bc=cb, bd=db, ebe=b-1, dcd-1=ab-1c, ece=a-1c-1, ede=d-1 >

Subgroups: 1060 in 85 conjugacy classes, 17 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, He3, C33, C33, C3×D9, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C32×C9, C3×He3, C32⋊D9, C3×C9⋊S3, He34S3, He3⋊C9, He32D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C322D9, C33⋊S3, He3.3S3, He3⋊S3, He32D9

Character table of He32D9

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 9O size 1 81 2 2 2 2 3 3 6 6 18 18 18 81 81 6 6 6 6 6 6 6 6 6 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 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 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 2 2 2 2 2 2 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 2 2 2 2 2 2 2 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 2 2 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 2 2 2 2 2 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 orthogonal lifted from S3 ρ7 2 0 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 0 0 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ8 2 0 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 0 0 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 0 -1 -1 -1 2 2 2 -1 -1 -1 2 -1 0 0 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 0 -1 -1 -1 2 2 2 -1 -1 2 -1 -1 0 0 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 2 0 0 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ12 2 0 -1 -1 -1 2 2 2 -1 -1 2 -1 -1 0 0 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ13 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 2 0 0 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ14 2 0 -1 -1 -1 2 2 2 -1 -1 2 -1 -1 0 0 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 2 0 0 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ16 3 -1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ17 3 1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ3 ζ32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ18 3 1 3 3 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ32 ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ19 3 -1 3 3 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from He3⋊C2 ρ20 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 -3 0 3 -3 0 3 -3 0 3 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ21 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 3 -3 0 3 -3 0 3 -3 0 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ22 6 0 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 3 -3 0 3 -3 0 3 -3 0 0 0 0 0 0 orthogonal lifted from C33⋊S3 ρ23 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 0 0 0 0 0 0 orthogonal lifted from He3⋊S3 ρ24 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ25 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ26 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 0 0 0 0 0 0 orthogonal lifted from He3⋊S3 ρ27 6 0 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 0 0 0 0 0 0 orthogonal lifted from He3⋊S3 ρ28 6 0 -3 6 -3 -3 0 0 0 0 0 0 0 0 0 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 2ζ95+ζ94+ζ92-ζ9 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ97-ζ94+2ζ92 ζ98+ζ94-ζ92+2ζ9 0 0 0 0 0 0 orthogonal lifted from He3.3S3 ρ29 6 0 -3 -3 -3 6 -3-3√-3 -3+3√-3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2D9 ρ30 6 0 -3 -3 -3 6 -3+3√-3 -3-3√-3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C32⋊2D9

Smallest permutation representation of He32D9
On 81 points
Generators in S81
(1 44 12)(2 45 13)(3 37 14)(4 38 15)(5 39 16)(6 40 17)(7 41 18)(8 42 10)(9 43 11)(19 66 49)(20 67 50)(21 68 51)(22 69 52)(23 70 53)(24 71 54)(25 72 46)(26 64 47)(27 65 48)(28 81 58)(29 73 59)(30 74 60)(31 75 61)(32 76 62)(33 77 63)(34 78 55)(35 79 56)(36 80 57)
(1 77 20)(2 78 21)(3 79 22)(4 80 23)(5 81 24)(6 73 25)(7 74 26)(8 75 27)(9 76 19)(10 31 48)(11 32 49)(12 33 50)(13 34 51)(14 35 52)(15 36 53)(16 28 54)(17 29 46)(18 30 47)(37 56 69)(38 57 70)(39 58 71)(40 59 72)(41 60 64)(42 61 65)(43 62 66)(44 63 67)(45 55 68)
(2 45 34)(3 14 69)(5 39 28)(6 17 72)(8 42 31)(9 11 66)(10 27 65)(12 50 33)(13 21 68)(15 53 36)(16 24 71)(18 47 30)(19 49 62)(22 52 56)(25 46 59)(29 40 73)(32 43 76)(35 37 79)(38 57 70)(41 60 64)(44 63 67)(48 75 61)(51 78 55)(54 81 58)
(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 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)(19 77)(20 76)(21 75)(22 74)(23 73)(24 81)(25 80)(26 79)(27 78)(37 64)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 53)(47 52)(48 51)(49 50)(55 61)(56 60)(57 59)(62 63)

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

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

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

Matrix representation of He32D9 in GL8(𝔽19)

 1 0 0 0 0 0 0 0 0 1 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 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 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
,
 18 1 0 0 0 0 0 0 18 0 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 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18
,
 12 2 0 0 0 0 0 0 17 14 0 0 0 0 0 0 0 0 1 4 3 18 1 4 0 0 15 16 1 4 15 16 0 0 1 4 1 4 3 18 0 0 15 16 15 16 1 4 0 0 3 18 1 4 1 4 0 0 1 4 15 16 15 16
,
 12 2 0 0 0 0 0 0 14 7 0 0 0 0 0 0 0 0 1 4 3 18 1 4 0 0 3 18 15 16 3 18 0 0 3 18 3 18 15 16 0 0 15 16 15 16 1 4 0 0 1 4 15 16 15 16 0 0 3 18 1 4 1 4

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[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],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[12,17,0,0,0,0,0,0,2,14,0,0,0,0,0,0,0,0,1,15,1,15,3,1,0,0,4,16,4,16,18,4,0,0,3,1,1,15,1,15,0,0,18,4,4,16,4,16,0,0,1,15,3,1,1,15,0,0,4,16,18,4,4,16],[12,14,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,1,3,3,15,1,3,0,0,4,18,18,16,4,18,0,0,3,15,3,15,15,1,0,0,18,16,18,16,16,4,0,0,1,3,15,1,15,1,0,0,4,18,16,4,16,4] >;

He32D9 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2D_9
% in TeX

G:=Group("He3:2D9");
// GroupNames label

G:=SmallGroup(486,56);
// by ID

G=gap.SmallGroup(486,56);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,1190,224,338,8211,2169,3244,11669]);
// 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=e*a*e=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a*b^-1*c,e*c*e=a^-1*c^-1,e*d*e=d^-1>;
// generators/relations

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