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G = He3⋊2D9  order 486 = 2·35

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

non-abelian, supersoluble, monomial

Aliases: He3⋊2D9, He3⋊C9⋊3C2, C32⋊C9⋊8S3, (C32×C9)⋊11S3, (C3×He3).6S3, C32.1(C9⋊S3), C33.23(C3⋊S3), C3.7(C33⋊S3), C3.5(He3⋊S3), C3.7(C32⋊2D9), 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 He3⋊2D9
 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, He3⋊4S3, He3⋊C9, He3⋊2D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C32⋊2D9, C33⋊S3, He3.3S3, He3⋊S3, He3⋊2D9

Character table of He3⋊2D9

 class 123A3B3C3D3E3F3G3H3I3J3K6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O
 size 181222233661818188181666666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-111111111111-1-1111111111111111    linear of order 2
ρ32022222222-1-1-100222222222-1-1-1-1-1-1    orthogonal lifted from S3
ρ42022222222-1-1-100-1-1-1-1-1-1-1-1-1222-1-1-1    orthogonal lifted from S3
ρ5202222222222200-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62022222222-1-1-100-1-1-1-1-1-1-1-1-1-1-1-1222    orthogonal lifted from S3
ρ720-1-1-1222-1-1-12-100ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ820-1-1-1222-1-1-12-100ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ920-1-1-1222-1-1-12-100ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ1020-1-1-1222-1-12-1-100ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ1120-1-1-1222-1-1-1-1200ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ1220-1-1-1222-1-12-1-100ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ1320-1-1-1222-1-1-1-1200ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ1420-1-1-1222-1-12-1-100ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ1520-1-1-1222-1-1-1-1200ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ163-13333-3+3√-3/2-3-3√-3/2-3+3√-3/2-3-3√-3/2000ζ6ζ65000000000000000    complex lifted from He3⋊C2
ρ17313333-3-3√-3/2-3+3√-3/2-3-3√-3/2-3+3√-3/2000ζ3ζ32000000000000000    complex lifted from He3⋊C2
ρ18313333-3+3√-3/2-3-3√-3/2-3+3√-3/2-3-3√-3/2000ζ32ζ3000000000000000    complex lifted from He3⋊C2
ρ193-13333-3-3√-3/2-3+3√-3/2-3-3√-3/2-3+3√-3/2000ζ65ζ6000000000000000    complex lifted from He3⋊C2
ρ20606-3-3-3000000000-303-303-303000000    orthogonal lifted from C33⋊S3
ρ21606-3-3-30000000003-303-303-30000000    orthogonal lifted from C33⋊S3
ρ22606-3-3-300000000003-303-303-3000000    orthogonal lifted from C33⋊S3
ρ2360-3-36-3000000000-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9000000    orthogonal lifted from He3⋊S3
ρ2460-36-3-3000000000ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92000000    orthogonal lifted from He3.3S3
ρ2560-36-3-30000000002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9000000    orthogonal lifted from He3.3S3
ρ2660-3-36-30000000002ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92000000    orthogonal lifted from He3⋊S3
ρ2760-3-36-3000000000ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ92ζ95+2ζ94-ζ92+ζ92ζ98-ζ94+ζ92+ζ9000000    orthogonal lifted from He3⋊S3
ρ2860-36-3-3000000000ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ9000000    orthogonal lifted from He3.3S3
ρ2960-3-3-36-3-3√-3-3+3√-33+3√-3/23-3√-3/200000000000000000000    complex lifted from C32⋊2D9
ρ3060-3-3-36-3+3√-3-3-3√-33-3√-3/23+3√-3/200000000000000000000    complex lifted from C32⋊2D9

Smallest permutation representation of He3⋊2D9
►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 He3⋊2D9 ►in GL8(𝔽19)

10000000
01000000
00001000
00000100
00000010
00000001
00100000
00010000
,
10000000
01000000
00010000
0018180000
00000100
0000181800
00000001
0000001818
,
181000000
180000000
00100000
00010000
0000181800
00001000
00000001
0000001818
,
122000000
1714000000
001431814
001516141516
001414318
001516151614
003181414
001415161516
,
122000000
147000000
001431814
003181516318
003183181516
001516151614
001415161516
003181414

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] >;
 

He3⋊2D9 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
 

Export

Character table of He3⋊2D9 in TeX

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