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G = He3.D9order 486 = 2·35

1st non-split extension by He3 of D9 acting via D9/C3=S3

metabelian, supersoluble, monomial

Aliases: He3.1D9, C27⋊S32C3, (C3×C27)⋊2C6, C9.1(C9⋊C6), C9○He3.1S3, C9.5He32C2, C32.2(C3×D9), C9.4(C32⋊C6), C3.3(C32⋊D9), (C3×C9).36(C3×S3), SmallGroup(486,27)

Series: Derived Chief Lower central Upper central

C1C3×C27 — He3.D9
C1C3C9C3×C9C3×C27C9.5He3 — He3.D9
C3×C27 — He3.D9
C1

Generators and relations for He3.D9
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=ebe=b-1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, dcd-1=ece=ab-1c, ede=bd8 >

81C2
3C3
9C3
27S3
81S3
81C6
3C32
6C9
9D9
9D9
9C3⋊S3
9D9
27C3×S3
23- 1+2
33- 1+2
3C27
33- 1+2
3C3×C9
6C27
3C9⋊S3
9C3×D9
9C32⋊C6
9C9⋊C6
9D27
9C9⋊C6
2C27⋊C3
3He3.4S3

Character table of He3.D9

 class 123A3B3C3D6A6B9A9B9C9D9E9F9G27A27B27C27D27E27F27G27H27I27J27K27L27M27N27O
 size 18126998181222661818666666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-11111-1-11111111111111111111111    linear of order 2
ρ31-111ζ3ζ32ζ65ζ611111ζ32ζ3111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ41111ζ32ζ3ζ32ζ311111ζ3ζ32111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ51-111ζ32ζ3ζ6ζ6511111ζ3ζ32111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ61111ζ3ζ32ζ3ζ3211111ζ32ζ3111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ7202222002222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ820222200-1-1-1-1-1-1-1ζ989ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ920222200-1-1-1-1-1-1-1ζ9792ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1020222200-1-1-1-1-1-1-1ζ9594ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ112022-1--3-1+-30022222-1+-3-1--3-1-1-1-1-1-1-1-1-1ζ6ζ6ζ6ζ65ζ65ζ65    complex lifted from C3×S3
ρ122022-1+-3-1--30022222-1--3-1+-3-1-1-1-1-1-1-1-1-1ζ65ζ65ζ65ζ6ζ6ζ6    complex lifted from C3×S3
ρ132022-1+-3-1--300-1-1-1-1-1ζ6ζ65ζ9792ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ959ζ9897ζ9492ζ9795ζ9894ζ929    complex lifted from C3×D9
ρ142022-1+-3-1--300-1-1-1-1-1ζ6ζ65ζ9594ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9897ζ9492ζ959ζ9894ζ929ζ9795    complex lifted from C3×D9
ρ152022-1+-3-1--300-1-1-1-1-1ζ6ζ65ζ989ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ9492ζ959ζ9897ζ929ζ9795ζ9894    complex lifted from C3×D9
ρ162022-1--3-1+-300-1-1-1-1-1ζ65ζ6ζ989ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ9795ζ9894ζ929ζ9897ζ9492ζ959    complex lifted from C3×D9
ρ172022-1--3-1+-300-1-1-1-1-1ζ65ζ6ζ9792ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9894ζ929ζ9795ζ9492ζ959ζ9897    complex lifted from C3×D9
ρ182022-1--3-1+-300-1-1-1-1-1ζ65ζ6ζ9594ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ929ζ9795ζ9894ζ959ζ9897ζ9492    complex lifted from C3×D9
ρ19606-30000666-3-300000000000000000    orthogonal lifted from C32⋊C6
ρ20606-30000-3-3-3-3600000000000000000    orthogonal lifted from C9⋊C6
ρ21606-30000-3-3-36-300000000000000000    orthogonal lifted from C9⋊C6
ρ2260-3000002724+3ζ2732715+3ζ27122721+3ζ2760000ζ2717+2ζ271027827ζ2714+2ζ271327527427262710278272726+2ζ2719271027827202716277272ζ27252711277+2ζ27227162711277272272327132752742723+2ζ27222713275000000    orthogonal faithful
ρ2360-3000002721+3ζ2762724+3ζ2732715+3ζ27120000ζ27252711277+2ζ2722726+2ζ271927102782716271127727227202716277272272327132752742723+2ζ27222713275ζ2714+2ζ2713275274ζ2717+2ζ2710278272726271027827000000    orthogonal faithful
ρ2460-3000002715+3ζ27122721+3ζ2762724+3ζ27300002723+2ζ2722271327527202716277272ζ2714+2ζ271327527427232713275274ζ2717+2ζ27102782727262710278272726+2ζ27192710278ζ27252711277+2ζ27227162711277272000000    orthogonal faithful
ρ2560-3000002715+3ζ27122721+3ζ2762724+3ζ2730000ζ2714+2ζ2713275274ζ27252711277+2ζ272272327132752742723+2ζ2722271327527262710278272726+2ζ27192710278ζ2717+2ζ2710278272716271127727227202716277272000000    orthogonal faithful
ρ2660-3000002724+3ζ2732715+3ζ27122721+3ζ27600002726+2ζ271927102782723+2ζ27222713275ζ2717+2ζ27102782727262710278272716271127727227202716277272ζ27252711277+2ζ272ζ2714+2ζ271327527427232713275274000000    orthogonal faithful
ρ2760-3000002724+3ζ2732715+3ζ27122721+3ζ27600002726271027827272327132752742726+2ζ27192710278ζ2717+2ζ271027827ζ27252711277+2ζ27227162711277272272027162772722723+2ζ27222713275ζ2714+2ζ2713275274000000    orthogonal faithful
ρ2860-3000002721+3ζ2762724+3ζ2732715+3ζ27120000272027162772722726271027827ζ27252711277+2ζ27227162711277272ζ2714+2ζ2713275274272327132752742723+2ζ272227132752726+2ζ27192710278ζ2717+2ζ271027827000000    orthogonal faithful
ρ2960-3000002721+3ζ2762724+3ζ2732715+3ζ2712000027162711277272ζ2717+2ζ27102782727202716277272ζ27252711277+2ζ2722723+2ζ27222713275ζ2714+2ζ27132752742723271327527427262710278272726+2ζ27192710278000000    orthogonal faithful
ρ3060-3000002715+3ζ27122721+3ζ2762724+3ζ273000027232713275274271627112772722723+2ζ27222713275ζ2714+2ζ27132752742726+2ζ27192710278ζ2717+2ζ271027827272627102782727202716277272ζ27252711277+2ζ272000000    orthogonal faithful

Smallest permutation representation of He3.D9
On 81 points
Generators in S81
(1 63 34)(2 64 35)(3 65 36)(4 66 37)(5 67 38)(6 68 39)(7 69 40)(8 70 41)(9 71 42)(10 72 43)(11 73 44)(12 74 45)(13 75 46)(14 76 47)(15 77 48)(16 78 49)(17 79 50)(18 80 51)(19 81 52)(20 55 53)(21 56 54)(22 57 28)(23 58 29)(24 59 30)(25 60 31)(26 61 32)(27 62 33)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(55 73 64)(56 74 65)(57 75 66)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)
(2 64 53)(3 36 74)(5 67 29)(6 39 77)(8 70 32)(9 42 80)(11 73 35)(12 45 56)(14 76 38)(15 48 59)(17 79 41)(18 51 62)(20 55 44)(21 54 65)(23 58 47)(24 30 68)(26 61 50)(27 33 71)(28 37 46)(31 40 49)(34 43 52)(57 75 66)(60 78 69)(63 81 72)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 27)(20 26)(21 25)(22 24)(28 59)(29 58)(30 57)(31 56)(32 55)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)

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

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

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

Matrix representation of He3.D9 in GL6(𝔽109)

001000
000100
000010
000001
100000
010000
,
10810000
10800000
00108100
00108000
00001081
00001080
,
100000
010000
00010800
00110800
00001081
00001080
,
18810018
101991081019
18188100
101910199108
81001818
910810191019
,
91081081019108
810010018100
10810191089108
100181008100
91089108108101
810081001001

G:=sub<GL(6,GF(109))| [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],[108,108,0,0,0,0,1,0,0,0,0,0,0,0,108,108,0,0,0,0,1,0,0,0,0,0,0,0,108,108,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,108,108,0,0,0,0,0,0,108,108,0,0,0,0,1,0],[1,101,1,101,8,9,8,9,8,9,100,108,8,9,1,101,1,101,100,108,8,9,8,9,1,101,8,9,1,101,8,9,100,108,8,9],[9,8,108,100,9,8,108,100,101,1,108,100,108,100,9,8,9,8,101,1,108,100,108,100,9,8,9,8,108,100,108,100,108,100,101,1] >;

He3.D9 in GAP, Magma, Sage, TeX

{\rm He}_3.D_9
% in TeX

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

G:=SmallGroup(486,27);
// by ID

G=gap.SmallGroup(486,27);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,1310,867,2169,12964,118,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^9=e*b*e=b^-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,d*c*d^-1=e*c*e=a*b^-1*c,e*d*e=b*d^8>;
// generators/relations

Export

Subgroup lattice of He3.D9 in TeX
Character table of He3.D9 in TeX

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