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## G = He3.5D9order 486 = 2·35

### The non-split extension by He3 of D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3.5D9
G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=b, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=b-1d8 >

Subgroups: 458 in 62 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C27, C27, C27, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, D27, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C27, C3×C27, C27⋊C3, C27⋊C3, C9○He3, C3×D27, C27⋊C6, C27⋊S3, He3.4S3, C27○He3, He3.5D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, He3.5D9

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

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

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

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

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D ··· 9K 27A ··· 27I 27J ··· 27AG order 1 2 3 3 3 3 3 3 6 6 9 9 9 9 ··· 9 27 ··· 27 27 ··· 27 size 1 81 2 3 3 6 6 6 81 81 2 2 2 6 ··· 6 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 6 type + + + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 C3×S3 D9 D9 C3×D9 C3×D9 He3.5D9 kernel He3.5D9 C27○He3 C27⋊S3 C3×C27 C3×C27 C27⋊C3 C9○He3 C27 C3×C9 He3 3- 1+2 C9 C32 C1 # reps 1 1 2 2 1 2 1 6 2 3 6 12 6 9

Matrix representation of He3.5D9 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
,
 0 1 0 0 0 0 108 108 0 0 0 0 0 0 0 1 0 0 0 0 108 108 0 0 0 0 0 0 0 1 0 0 0 0 108 108
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 108 108 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 108 108
,
 92 99 0 0 0 0 10 102 0 0 0 0 0 0 92 99 0 0 0 0 10 102 0 0 0 0 0 0 92 99 0 0 0 0 10 102
,
 87 29 0 0 0 0 51 22 0 0 0 0 0 0 0 0 87 29 0 0 0 0 51 22 0 0 87 29 0 0 0 0 51 22 0 0

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],[0,108,0,0,0,0,1,108,0,0,0,0,0,0,0,108,0,0,0,0,1,108,0,0,0,0,0,0,0,108,0,0,0,0,1,108],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,108,1,0,0,0,0,108,0,0,0,0,0,0,0,0,108,0,0,0,0,1,108],[92,10,0,0,0,0,99,102,0,0,0,0,0,0,92,10,0,0,0,0,99,102,0,0,0,0,0,0,92,10,0,0,0,0,99,102],[87,51,0,0,0,0,29,22,0,0,0,0,0,0,0,0,87,51,0,0,0,0,29,22,0,0,87,51,0,0,0,0,29,22,0,0] >;

He3.5D9 in GAP, Magma, Sage, TeX

{\rm He}_3._5D_9
% in TeX

G:=Group("He3.5D9");
// GroupNames label

G:=SmallGroup(486,163);
// by ID

G=gap.SmallGroup(486,163);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,824,867,2169,8104,208,11669]);
// Polycyclic

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

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