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## G = He3.D9order 486 = 2·35

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

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

 Derived series C1 — C3×C27 — He3.D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C9.5He3 — He3.D9
 Lower central C3×C27 — He3.D9
 Upper central 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 >

Character table of He3.D9

 class 1 2 3A 3B 3C 3D 6A 6B 9A 9B 9C 9D 9E 9F 9G 27A 27B 27C 27D 27E 27F 27G 27H 27I 27J 27K 27L 27M 27N 27O size 1 81 2 6 9 9 81 81 2 2 2 6 6 18 18 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 1 -1 1 1 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ5 1 -1 1 1 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ7 2 0 2 2 2 2 0 0 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ9 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 0 2 2 -1-√-3 -1+√-3 0 0 2 2 2 2 2 -1+√-3 -1-√-3 -1 -1 -1 -1 -1 -1 -1 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 complex lifted from C3×S3 ρ12 2 0 2 2 -1+√-3 -1-√-3 0 0 2 2 2 2 2 -1-√-3 -1+√-3 -1 -1 -1 -1 -1 -1 -1 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 complex lifted from C3×S3 ρ13 2 0 2 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 -1 -1 ζ6 ζ65 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ9 ζ98+ζ97 ζ94+ζ92 ζ97+ζ95 ζ98+ζ94 ζ92+ζ9 complex lifted from C3×D9 ρ14 2 0 2 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 -1 -1 ζ6 ζ65 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ97 ζ94+ζ92 ζ95+ζ9 ζ98+ζ94 ζ92+ζ9 ζ97+ζ95 complex lifted from C3×D9 ρ15 2 0 2 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 -1 -1 ζ6 ζ65 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ94+ζ92 ζ95+ζ9 ζ98+ζ97 ζ92+ζ9 ζ97+ζ95 ζ98+ζ94 complex lifted from C3×D9 ρ16 2 0 2 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 -1 -1 ζ65 ζ6 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ95 ζ98+ζ94 ζ92+ζ9 ζ98+ζ97 ζ94+ζ92 ζ95+ζ9 complex lifted from C3×D9 ρ17 2 0 2 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 -1 -1 ζ65 ζ6 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ94 ζ92+ζ9 ζ97+ζ95 ζ94+ζ92 ζ95+ζ9 ζ98+ζ97 complex lifted from C3×D9 ρ18 2 0 2 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 -1 -1 ζ65 ζ6 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ92+ζ9 ζ97+ζ95 ζ98+ζ94 ζ95+ζ9 ζ98+ζ97 ζ94+ζ92 complex lifted from C3×D9 ρ19 6 0 6 -3 0 0 0 0 6 6 6 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 0 6 -3 0 0 0 0 -3 -3 -3 -3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ21 6 0 6 -3 0 0 0 0 -3 -3 -3 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ22 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2715+3ζ2712 3ζ2721+3ζ276 0 0 0 0 ζ2717+2ζ2710-ζ278+ζ27 ζ2714+2ζ2713-ζ275+ζ274 2ζ2726-ζ2710+ζ278+ζ27 -ζ2726+2ζ2719+ζ2710+ζ278 2ζ2720-ζ2716+ζ277+ζ272 ζ2725+ζ2711-ζ277+2ζ272 2ζ2716+ζ2711+ζ277-ζ272 2ζ2723-ζ2713+ζ275+ζ274 -ζ2723+2ζ2722+ζ2713+ζ275 0 0 0 0 0 0 orthogonal faithful ρ23 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2724+3ζ273 3ζ2715+3ζ2712 0 0 0 0 ζ2725+ζ2711-ζ277+2ζ272 -ζ2726+2ζ2719+ζ2710+ζ278 2ζ2716+ζ2711+ζ277-ζ272 2ζ2720-ζ2716+ζ277+ζ272 2ζ2723-ζ2713+ζ275+ζ274 -ζ2723+2ζ2722+ζ2713+ζ275 ζ2714+2ζ2713-ζ275+ζ274 ζ2717+2ζ2710-ζ278+ζ27 2ζ2726-ζ2710+ζ278+ζ27 0 0 0 0 0 0 orthogonal faithful ρ24 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2721+3ζ276 3ζ2724+3ζ273 0 0 0 0 -ζ2723+2ζ2722+ζ2713+ζ275 2ζ2720-ζ2716+ζ277+ζ272 ζ2714+2ζ2713-ζ275+ζ274 2ζ2723-ζ2713+ζ275+ζ274 ζ2717+2ζ2710-ζ278+ζ27 2ζ2726-ζ2710+ζ278+ζ27 -ζ2726+2ζ2719+ζ2710+ζ278 ζ2725+ζ2711-ζ277+2ζ272 2ζ2716+ζ2711+ζ277-ζ272 0 0 0 0 0 0 orthogonal faithful ρ25 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2721+3ζ276 3ζ2724+3ζ273 0 0 0 0 ζ2714+2ζ2713-ζ275+ζ274 ζ2725+ζ2711-ζ277+2ζ272 2ζ2723-ζ2713+ζ275+ζ274 -ζ2723+2ζ2722+ζ2713+ζ275 2ζ2726-ζ2710+ζ278+ζ27 -ζ2726+2ζ2719+ζ2710+ζ278 ζ2717+2ζ2710-ζ278+ζ27 2ζ2716+ζ2711+ζ277-ζ272 2ζ2720-ζ2716+ζ277+ζ272 0 0 0 0 0 0 orthogonal faithful ρ26 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2715+3ζ2712 3ζ2721+3ζ276 0 0 0 0 -ζ2726+2ζ2719+ζ2710+ζ278 -ζ2723+2ζ2722+ζ2713+ζ275 ζ2717+2ζ2710-ζ278+ζ27 2ζ2726-ζ2710+ζ278+ζ27 2ζ2716+ζ2711+ζ277-ζ272 2ζ2720-ζ2716+ζ277+ζ272 ζ2725+ζ2711-ζ277+2ζ272 ζ2714+2ζ2713-ζ275+ζ274 2ζ2723-ζ2713+ζ275+ζ274 0 0 0 0 0 0 orthogonal faithful ρ27 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2715+3ζ2712 3ζ2721+3ζ276 0 0 0 0 2ζ2726-ζ2710+ζ278+ζ27 2ζ2723-ζ2713+ζ275+ζ274 -ζ2726+2ζ2719+ζ2710+ζ278 ζ2717+2ζ2710-ζ278+ζ27 ζ2725+ζ2711-ζ277+2ζ272 2ζ2716+ζ2711+ζ277-ζ272 2ζ2720-ζ2716+ζ277+ζ272 -ζ2723+2ζ2722+ζ2713+ζ275 ζ2714+2ζ2713-ζ275+ζ274 0 0 0 0 0 0 orthogonal faithful ρ28 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2724+3ζ273 3ζ2715+3ζ2712 0 0 0 0 2ζ2720-ζ2716+ζ277+ζ272 2ζ2726-ζ2710+ζ278+ζ27 ζ2725+ζ2711-ζ277+2ζ272 2ζ2716+ζ2711+ζ277-ζ272 ζ2714+2ζ2713-ζ275+ζ274 2ζ2723-ζ2713+ζ275+ζ274 -ζ2723+2ζ2722+ζ2713+ζ275 -ζ2726+2ζ2719+ζ2710+ζ278 ζ2717+2ζ2710-ζ278+ζ27 0 0 0 0 0 0 orthogonal faithful ρ29 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2724+3ζ273 3ζ2715+3ζ2712 0 0 0 0 2ζ2716+ζ2711+ζ277-ζ272 ζ2717+2ζ2710-ζ278+ζ27 2ζ2720-ζ2716+ζ277+ζ272 ζ2725+ζ2711-ζ277+2ζ272 -ζ2723+2ζ2722+ζ2713+ζ275 ζ2714+2ζ2713-ζ275+ζ274 2ζ2723-ζ2713+ζ275+ζ274 2ζ2726-ζ2710+ζ278+ζ27 -ζ2726+2ζ2719+ζ2710+ζ278 0 0 0 0 0 0 orthogonal faithful ρ30 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2721+3ζ276 3ζ2724+3ζ273 0 0 0 0 2ζ2723-ζ2713+ζ275+ζ274 2ζ2716+ζ2711+ζ277-ζ272 -ζ2723+2ζ2722+ζ2713+ζ275 ζ2714+2ζ2713-ζ275+ζ274 -ζ2726+2ζ2719+ζ2710+ζ278 ζ2717+2ζ2710-ζ278+ζ27 2ζ2726-ζ2710+ζ278+ζ27 2ζ2720-ζ2716+ζ277+ζ272 ζ2725+ζ2711-ζ277+2ζ272 0 0 0 0 0 0 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)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 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 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
,
 1 8 8 100 1 8 101 9 9 108 101 9 1 8 1 8 8 100 101 9 101 9 9 108 8 100 1 8 1 8 9 108 101 9 101 9
,
 9 108 108 101 9 108 8 100 100 1 8 100 108 101 9 108 9 108 100 1 8 100 8 100 9 108 9 108 108 101 8 100 8 100 100 1

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

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