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## G = He3.4D9order 486 = 2·35

### 4th non-split extension by He3 of D9 acting via D9/C3=S3

Aliases: He3.4D9, 3- 1+2.2D9, C27⋊C33S3, (C3×C27)⋊6S3, C9○He3.4S3, C9.6He31C2, C32.4(C9⋊S3), C9.4(He3⋊C2), C3.10(C322D9), (C3×C9).11(C3⋊S3), SmallGroup(486,59)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9.6He3 — He3.4D9
 Chief series C1 — C3 — C32 — C3×C9 — C9○He3 — C9.6He3 — He3.4D9
 Lower central C9.6He3 — He3.4D9
 Upper central C1

Generators and relations for He3.4D9
G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=ebe=b-1, dad-1=eae=ab=ba, cac-1=ab-1, bc=cb, bd=db, dcd-1=a-1c, ece=abc-1, ede=bd8 >

Character table of He3.4D9

 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 3 3 18 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 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 ρ4 2 0 2 2 2 -1 0 0 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 -1 2 -1 orthogonal lifted from S3 ρ5 2 0 2 2 2 -1 0 0 2 2 2 2 2 -1 -1 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 2 2 2 -1 0 0 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 2 -1 2 orthogonal lifted from S3 ρ7 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 2 -1 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ8 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 -1 2 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 -1 2 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 -1 2 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 2 -1 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ12 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ13 2 0 2 2 2 -1 0 0 -1 -1 -1 -1 -1 2 -1 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ14 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ16 3 -1 3 -3+3√-3/2 -3-3√-3/2 0 ζ65 ζ6 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 complex lifted from He3⋊C2 ρ17 3 -1 3 -3-3√-3/2 -3+3√-3/2 0 ζ6 ζ65 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 complex lifted from He3⋊C2 ρ18 3 1 3 -3-3√-3/2 -3+3√-3/2 0 ζ32 ζ3 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 complex lifted from He3⋊C2 ρ19 3 1 3 -3+3√-3/2 -3-3√-3/2 0 ζ3 ζ32 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 complex lifted from He3⋊C2 ρ20 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2721+3ζ276 3ζ2715+3ζ2712 0 0 0 0 2ζ2714+ζ2713+ζ275-ζ274 ζ2723+ζ2722-ζ2713+2ζ275 2ζ2717+ζ2710+ζ278-ζ27 ζ2726+ζ2719-ζ2710+2ζ278 ζ2726+ζ2710-ζ278+2ζ27 ζ2716+2ζ2711-ζ277+ζ272 2ζ2725-ζ2711+ζ277+ζ272 ζ2720+ζ2716+2ζ277-ζ272 ζ2723+ζ2713-ζ275+2ζ274 0 0 0 0 0 0 orthogonal faithful ρ21 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2724+3ζ273 3ζ2721+3ζ276 0 0 0 0 ζ2720+ζ2716+2ζ277-ζ272 ζ2716+2ζ2711-ζ277+ζ272 ζ2723+ζ2722-ζ2713+2ζ275 ζ2723+ζ2713-ζ275+2ζ274 2ζ2714+ζ2713+ζ275-ζ274 ζ2726+ζ2719-ζ2710+2ζ278 ζ2726+ζ2710-ζ278+2ζ27 2ζ2717+ζ2710+ζ278-ζ27 2ζ2725-ζ2711+ζ277+ζ272 0 0 0 0 0 0 orthogonal faithful ρ22 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2715+3ζ2712 3ζ2724+3ζ273 0 0 0 0 2ζ2717+ζ2710+ζ278-ζ27 ζ2726+ζ2719-ζ2710+2ζ278 ζ2716+2ζ2711-ζ277+ζ272 2ζ2725-ζ2711+ζ277+ζ272 ζ2720+ζ2716+2ζ277-ζ272 ζ2723+ζ2713-ζ275+2ζ274 2ζ2714+ζ2713+ζ275-ζ274 ζ2723+ζ2722-ζ2713+2ζ275 ζ2726+ζ2710-ζ278+2ζ27 0 0 0 0 0 0 orthogonal faithful ρ23 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2721+3ζ276 3ζ2715+3ζ2712 0 0 0 0 ζ2723+ζ2713-ζ275+2ζ274 2ζ2714+ζ2713+ζ275-ζ274 ζ2726+ζ2710-ζ278+2ζ27 2ζ2717+ζ2710+ζ278-ζ27 ζ2726+ζ2719-ζ2710+2ζ278 ζ2720+ζ2716+2ζ277-ζ272 ζ2716+2ζ2711-ζ277+ζ272 2ζ2725-ζ2711+ζ277+ζ272 ζ2723+ζ2722-ζ2713+2ζ275 0 0 0 0 0 0 orthogonal faithful ρ24 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2724+3ζ273 3ζ2721+3ζ276 0 0 0 0 2ζ2725-ζ2711+ζ277+ζ272 ζ2720+ζ2716+2ζ277-ζ272 2ζ2714+ζ2713+ζ275-ζ274 ζ2723+ζ2722-ζ2713+2ζ275 ζ2723+ζ2713-ζ275+2ζ274 2ζ2717+ζ2710+ζ278-ζ27 ζ2726+ζ2719-ζ2710+2ζ278 ζ2726+ζ2710-ζ278+2ζ27 ζ2716+2ζ2711-ζ277+ζ272 0 0 0 0 0 0 orthogonal faithful ρ25 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2715+3ζ2712 3ζ2724+3ζ273 0 0 0 0 ζ2726+ζ2719-ζ2710+2ζ278 ζ2726+ζ2710-ζ278+2ζ27 2ζ2725-ζ2711+ζ277+ζ272 ζ2720+ζ2716+2ζ277-ζ272 ζ2716+2ζ2711-ζ277+ζ272 2ζ2714+ζ2713+ζ275-ζ274 ζ2723+ζ2722-ζ2713+2ζ275 ζ2723+ζ2713-ζ275+2ζ274 2ζ2717+ζ2710+ζ278-ζ27 0 0 0 0 0 0 orthogonal faithful ρ26 6 0 -3 0 0 0 0 0 3ζ2721+3ζ276 3ζ2715+3ζ2712 3ζ2724+3ζ273 0 0 0 0 ζ2726+ζ2710-ζ278+2ζ27 2ζ2717+ζ2710+ζ278-ζ27 ζ2720+ζ2716+2ζ277-ζ272 ζ2716+2ζ2711-ζ277+ζ272 2ζ2725-ζ2711+ζ277+ζ272 ζ2723+ζ2722-ζ2713+2ζ275 ζ2723+ζ2713-ζ275+2ζ274 2ζ2714+ζ2713+ζ275-ζ274 ζ2726+ζ2719-ζ2710+2ζ278 0 0 0 0 0 0 orthogonal faithful ρ27 6 0 -3 0 0 0 0 0 3ζ2715+3ζ2712 3ζ2724+3ζ273 3ζ2721+3ζ276 0 0 0 0 ζ2716+2ζ2711-ζ277+ζ272 2ζ2725-ζ2711+ζ277+ζ272 ζ2723+ζ2713-ζ275+2ζ274 2ζ2714+ζ2713+ζ275-ζ274 ζ2723+ζ2722-ζ2713+2ζ275 ζ2726+ζ2710-ζ278+2ζ27 2ζ2717+ζ2710+ζ278-ζ27 ζ2726+ζ2719-ζ2710+2ζ278 ζ2720+ζ2716+2ζ277-ζ272 0 0 0 0 0 0 orthogonal faithful ρ28 6 0 -3 0 0 0 0 0 3ζ2724+3ζ273 3ζ2721+3ζ276 3ζ2715+3ζ2712 0 0 0 0 ζ2723+ζ2722-ζ2713+2ζ275 ζ2723+ζ2713-ζ275+2ζ274 ζ2726+ζ2719-ζ2710+2ζ278 ζ2726+ζ2710-ζ278+2ζ27 2ζ2717+ζ2710+ζ278-ζ27 2ζ2725-ζ2711+ζ277+ζ272 ζ2720+ζ2716+2ζ277-ζ272 ζ2716+2ζ2711-ζ277+ζ272 2ζ2714+ζ2713+ζ275-ζ274 0 0 0 0 0 0 orthogonal faithful ρ29 6 0 6 -3-3√-3 -3+3√-3 0 0 0 -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 complex lifted from C32⋊2D9 ρ30 6 0 6 -3+3√-3 -3-3√-3 0 0 0 -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 complex lifted from C32⋊2D9

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

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

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

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

Matrix representation of He3.4D9 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
,
 108 108 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
,
 98 41 52 11 98 41 68 57 98 41 68 57 52 11 52 11 68 57 98 41 98 41 52 11 98 41 68 57 68 57 68 57 52 11 52 11
,
 68 57 98 41 68 57 98 41 52 11 98 41 98 41 98 41 52 11 52 11 52 11 68 57 68 57 52 11 52 11 98 41 68 57 68 57

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],[98,68,52,98,98,68,41,57,11,41,41,57,52,98,52,98,68,52,11,41,11,41,57,11,98,68,68,52,68,52,41,57,57,11,57,11],[68,98,98,52,68,98,57,41,41,11,57,41,98,52,98,52,52,68,41,11,41,11,11,57,68,98,52,68,52,68,57,41,11,57,11,57] >;

He3.4D9 in GAP, Magma, Sage, TeX

{\rm He}_3._4D_9
% in TeX

G:=Group("He3.4D9");
// GroupNames label

G:=SmallGroup(486,59);
// by ID

G=gap.SmallGroup(486,59);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,1195,218,548,4755,453,3250,1906,11669]);
// Polycyclic

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

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