Copied to
clipboard

G = He3.3D9  order 486 = 2·35

3rd non-split extension by He3 of D9 acting via D9/C3=S3

non-abelian, supersoluble, monomial

Aliases: He3.3D9, 3- 1+2.1D9, C27⋊C3⋊2S3, (C3×C27)⋊5S3, C9○He3.3S3, C9.5He3⋊1C2, C32.3(C9⋊S3), C9.3(He3⋊C2), C3.9(C32⋊2D9), (C3×C9).10(C3⋊S3), SmallGroup(486,58)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C9.5He3 — He3.3D9
Chief series
C1 — C3 — C32 — C3×C9 — C9○He3 — C9.5He3 — He3.3D9
Lower central
C9.5He3 — He3.3D9
Upper central
C1

Generators and relations for He3.3D9
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=b, dad-1=eae=ab=ba, cac-1=ab-1, bc=cb, bd=db, ebe=b-1, dcd-1=a-1c, ece=abc-1, ede=b-1d8 >

C1
81C2
C3
3C3
9C3
27S3
81S3
81C6
C9
C32
2C9
3C32
3C9
3C9
9D9
9D9
9C3⋊S3
9D9
27C3×S3
3- 1+2
He3
3- 1+2
C3×C9
3C3×C9
3C27
3C27
3C27
63- 1+2
3D27
3D27
3C9⋊S3
3D27
9C9⋊C6
9C3×D9
9C9⋊C6
9C32⋊C6
C9○He3
C3×C27
C27⋊C3
C27⋊C3
3C27⋊C6
3He3.4S3
3C3×D27
3C27⋊C6
C9.5He3
He3.3D9

Character table of He3.3D9

 class 123A3B3C3D6A6B9A9B9C9D9E9F9G27A27B27C27D27E27F27G27H27I27J27K27L27M27N27O
 size 181233188181222661818666666666181818181818
ρ1111111111111111111111111111111    trivial
ρ21-11111-1-11111111111111111111111    linear of order 2
ρ3202222002222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ420222-10022222-1-1-1-1-1-1-1-1-1-1-1-122-12-1    orthogonal lifted from S3
ρ520222-10022222-1-1222222222-1-1-1-1-1-1    orthogonal lifted from S3
ρ620222-10022222-1-1-1-1-1-1-1-1-1-1-12-1-12-12    orthogonal lifted from S3
ρ720222-100-1-1-1-1-12-1ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ820222-100-1-1-1-1-1-12ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9    orthogonal lifted from D9
ρ920222-100-1-1-1-1-1-12ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ95+ζ94    orthogonal lifted from D9
ρ1020222-100-1-1-1-1-1-12ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92    orthogonal lifted from D9
ρ1120222-100-1-1-1-1-12-1ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ1220222200-1-1-1-1-1-1-1ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9    orthogonal lifted from D9
ρ1320222-100-1-1-1-1-12-1ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ1420222200-1-1-1-1-1-1-1ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92    orthogonal lifted from D9
ρ1520222200-1-1-1-1-1-1-1ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94    orthogonal lifted from D9
ρ163-13-3+3√-3/2-3-3√-3/20ζ65ζ6333-3+3√-3/2-3-3√-3/200000000000000000    complex lifted from He3⋊C2
ρ173-13-3-3√-3/2-3+3√-3/20ζ6ζ65333-3-3√-3/2-3+3√-3/200000000000000000    complex lifted from He3⋊C2
ρ18313-3-3√-3/2-3+3√-3/20ζ32ζ3333-3-3√-3/2-3+3√-3/200000000000000000    complex lifted from He3⋊C2
ρ19313-3+3√-3/2-3-3√-3/20ζ3ζ32333-3+3√-3/2-3-3√-3/200000000000000000    complex lifted from He3⋊C2
ρ2060-3000003ζ2715+3ζ27123ζ2721+3ζ2763ζ2724+3ζ27300002ζ2726-ζ2710+ζ278+ζ27ζ2719+ζ2717+2ζ278-ζ27-ζ2725+ζ2720+ζ2716+2ζ2711ζ2725+ζ2720-ζ2711+2ζ2772ζ2725-ζ2720+ζ2716+ζ272-ζ2723+2ζ2722+ζ2713+ζ275ζ2714+2ζ2713-ζ275+ζ2742ζ2723-ζ2713+ζ275+ζ274ζ2717+2ζ2710-ζ278+ζ27000000    orthogonal faithful
ρ2160-3000003ζ2715+3ζ27123ζ2721+3ζ2763ζ2724+3ζ2730000ζ2719+ζ2717+2ζ278-ζ27ζ2717+2ζ2710-ζ278+ζ27ζ2725+ζ2720-ζ2711+2ζ2772ζ2725-ζ2720+ζ2716+ζ272-ζ2725+ζ2720+ζ2716+2ζ2711ζ2714+2ζ2713-ζ275+ζ2742ζ2723-ζ2713+ζ275+ζ274-ζ2723+2ζ2722+ζ2713+ζ2752ζ2726-ζ2710+ζ278+ζ27000000    orthogonal faithful
ρ2260-3000003ζ2721+3ζ2763ζ2724+3ζ2733ζ2715+3ζ27120000ζ2714+2ζ2713-ζ275+ζ2742ζ2723-ζ2713+ζ275+ζ274ζ2719+ζ2717+2ζ278-ζ27ζ2717+2ζ2710-ζ278+ζ272ζ2726-ζ2710+ζ278+ζ27-ζ2725+ζ2720+ζ2716+2ζ2711ζ2725+ζ2720-ζ2711+2ζ2772ζ2725-ζ2720+ζ2716+ζ272-ζ2723+2ζ2722+ζ2713+ζ275000000    orthogonal faithful
ρ2360-3000003ζ2724+3ζ2733ζ2715+3ζ27123ζ2721+3ζ2760000-ζ2725+ζ2720+ζ2716+2ζ2711ζ2725+ζ2720-ζ2711+2ζ277ζ2714+2ζ2713-ζ275+ζ2742ζ2723-ζ2713+ζ275+ζ274-ζ2723+2ζ2722+ζ2713+ζ2752ζ2726-ζ2710+ζ278+ζ27ζ2719+ζ2717+2ζ278-ζ27ζ2717+2ζ2710-ζ278+ζ272ζ2725-ζ2720+ζ2716+ζ272000000    orthogonal faithful
ρ2460-3000003ζ2721+3ζ2763ζ2724+3ζ2733ζ2715+3ζ27120000-ζ2723+2ζ2722+ζ2713+ζ275ζ2714+2ζ2713-ζ275+ζ2742ζ2726-ζ2710+ζ278+ζ27ζ2719+ζ2717+2ζ278-ζ27ζ2717+2ζ2710-ζ278+ζ272ζ2725-ζ2720+ζ2716+ζ272-ζ2725+ζ2720+ζ2716+2ζ2711ζ2725+ζ2720-ζ2711+2ζ2772ζ2723-ζ2713+ζ275+ζ274000000    orthogonal faithful
ρ2560-3000003ζ2724+3ζ2733ζ2715+3ζ27123ζ2721+3ζ2760000ζ2725+ζ2720-ζ2711+2ζ2772ζ2725-ζ2720+ζ2716+ζ2722ζ2723-ζ2713+ζ275+ζ274-ζ2723+2ζ2722+ζ2713+ζ275ζ2714+2ζ2713-ζ275+ζ274ζ2719+ζ2717+2ζ278-ζ27ζ2717+2ζ2710-ζ278+ζ272ζ2726-ζ2710+ζ278+ζ27-ζ2725+ζ2720+ζ2716+2ζ2711000000    orthogonal faithful
ρ2660-3000003ζ2715+3ζ27123ζ2721+3ζ2763ζ2724+3ζ2730000ζ2717+2ζ2710-ζ278+ζ272ζ2726-ζ2710+ζ278+ζ272ζ2725-ζ2720+ζ2716+ζ272-ζ2725+ζ2720+ζ2716+2ζ2711ζ2725+ζ2720-ζ2711+2ζ2772ζ2723-ζ2713+ζ275+ζ274-ζ2723+2ζ2722+ζ2713+ζ275ζ2714+2ζ2713-ζ275+ζ274ζ2719+ζ2717+2ζ278-ζ27000000    orthogonal faithful
ρ2760-3000003ζ2724+3ζ2733ζ2715+3ζ27123ζ2721+3ζ27600002ζ2725-ζ2720+ζ2716+ζ272-ζ2725+ζ2720+ζ2716+2ζ2711-ζ2723+2ζ2722+ζ2713+ζ275ζ2714+2ζ2713-ζ275+ζ2742ζ2723-ζ2713+ζ275+ζ274ζ2717+2ζ2710-ζ278+ζ272ζ2726-ζ2710+ζ278+ζ27ζ2719+ζ2717+2ζ278-ζ27ζ2725+ζ2720-ζ2711+2ζ277000000    orthogonal faithful
ρ2860-3000003ζ2721+3ζ2763ζ2724+3ζ2733ζ2715+3ζ271200002ζ2723-ζ2713+ζ275+ζ274-ζ2723+2ζ2722+ζ2713+ζ275ζ2717+2ζ2710-ζ278+ζ272ζ2726-ζ2710+ζ278+ζ27ζ2719+ζ2717+2ζ278-ζ27ζ2725+ζ2720-ζ2711+2ζ2772ζ2725-ζ2720+ζ2716+ζ272-ζ2725+ζ2720+ζ2716+2ζ2711ζ2714+2ζ2713-ζ275+ζ274000000    orthogonal faithful
ρ29606-3-3√-3-3+3√-3000-3-3-33+3√-3/23-3√-3/200000000000000000    complex lifted from C32⋊2D9
ρ30606-3+3√-3-3-3√-3000-3-3-33-3√-3/23+3√-3/200000000000000000    complex lifted from C32⋊2D9

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

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

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

Matrix representation of He3.3D9 ►in GL6(𝔽109)

001000
000100
000010
000001
100000
010000
,
1081080000
100000
0010810800
001000
0000108108
000010
,
100000
010000
000100
0010810800
0000108108
000010
,
751246347512
976375129763
463446349763
751275124634
751297639763
976346344634
,
463497634634
976375129763
976397637512
751275124634
463475127512
976346344634

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],[108,1,0,0,0,0,108,0,0,0,0,0,0,0,108,1,0,0,0,0,108,0,0,0,0,0,0,0,108,1,0,0,0,0,108,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,108,0,0,0,0,1,108,0,0,0,0,0,0,108,1,0,0,0,0,108,0],[75,97,46,75,75,97,12,63,34,12,12,63,46,75,46,75,97,46,34,12,34,12,63,34,75,97,97,46,97,46,12,63,63,34,63,34],[46,97,97,75,46,97,34,63,63,12,34,63,97,75,97,75,75,46,63,12,63,12,12,34,46,97,75,46,75,46,34,63,12,34,12,34] >;
 

He3.3D9 in GAP, Magma, Sage, TeX 

{\rm He}_3._3D_9
 
% in TeX
 
G:=Group("He3.3D9");
 
// GroupNames label
 
G:=SmallGroup(486,58);
 
// by ID
 
G=gap.SmallGroup(486,58);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,2167,218,548,8643,237,3250,1906,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^9=b,d*a*d^-1=e*a*e=a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*c,e*c*e=a*b*c^-1,e*d*e=b^-1*d^8>;
 
// generators/relations
 

Export

Subgroup lattice of He3.3D9 in TeX
Character table of He3.3D9 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁