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## G = C33.5D9order 486 = 2·35

### 5th non-split extension by C33 of D9 acting via D9/C3=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C27 — C33.5D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C3×C27⋊C3 — C33.5D9
 Lower central C3×C27 — C33.5D9
 Upper central C1

Generators and relations for C33.5D9
G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=c, ab=ba, dad-1=eae=ac=ca, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=c-1d8 >

Subgroups: 548 in 72 conjugacy classes, 26 normal (13 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C27, C27, C3×C9, C3×C9, C3×C9, C33, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C27, C3×C27, C27⋊C3, C27⋊C3, C32×C9, C27⋊C6, C27⋊S3, C3×C9⋊S3, C3×C27⋊C3, C33.5D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C27⋊C6, C3×C9⋊S3, C33.5D9

Smallest permutation representation of C33.5D9
On 81 points
Generators in S81
```(2 11 20)(3 21 12)(5 14 23)(6 24 15)(8 17 26)(9 27 18)(28 37 46)(29 47 38)(31 40 49)(32 50 41)(34 43 52)(35 53 44)(55 64 73)(56 74 65)(58 67 76)(59 77 68)(61 70 79)(62 80 71)
(1 42 63)(2 43 64)(3 44 65)(4 45 66)(5 46 67)(6 47 68)(7 48 69)(8 49 70)(9 50 71)(10 51 72)(11 52 73)(12 53 74)(13 54 75)(14 28 76)(15 29 77)(16 30 78)(17 31 79)(18 32 80)(19 33 81)(20 34 55)(21 35 56)(22 36 57)(23 37 58)(24 38 59)(25 39 60)(26 40 61)(27 41 62)
(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)
(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 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)```

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

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

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

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9I 9J ··· 9O 27A ··· 27AA order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 27 ··· 27 size 1 81 2 2 2 2 3 3 6 6 81 81 2 ··· 2 6 ··· 6 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 6 type + + + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 D9 C3×D9 C3×D9 C27⋊C6 kernel C33.5D9 C3×C27⋊C3 C27⋊S3 C3×C27 C27⋊C3 C32×C9 C27 C3×C9 C3×C9 C33 C9 C32 C3 # reps 1 1 2 2 3 1 6 6 2 3 12 6 9

Matrix representation of C33.5D9 in GL8(𝔽109)

 45 0 0 0 0 0 0 0 0 45 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 74 74 108 108 0 0 0 0 74 74 1 0 0 0 0 0 42 42 0 0 0 1 0 0 24 24 0 0 108 108
,
 41 9 0 0 0 0 0 0 75 67 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 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 108 1 0 0 0 0 0 0 108 0 0 0 0 0 0 0 35 35 0 1 0 0 0 0 35 39 108 108 0 0 0 0 43 42 0 0 0 1 0 0 42 24 0 0 108 108
,
 41 9 0 0 0 0 0 0 75 67 0 0 0 0 0 0 0 0 35 35 108 1 0 0 0 0 74 74 107 108 0 0 0 0 73 73 0 0 1 0 0 0 6 6 105 0 0 1 0 0 71 44 1 0 0 0 0 0 103 71 18 0 0 0
,
 68 100 0 0 0 0 0 0 5 41 0 0 0 0 0 0 0 0 67 67 104 86 0 0 0 0 24 24 86 91 0 0 0 0 6 65 0 0 0 0 0 0 41 14 108 18 0 0 0 0 6 6 82 50 50 82 0 0 73 73 59 28 32 59

`G:=sub<GL(8,GF(109))| [45,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,1,0,74,74,42,24,0,0,0,1,74,74,42,24,0,0,0,0,108,1,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108],[41,75,0,0,0,0,0,0,9,67,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,108,108,35,35,43,42,0,0,1,0,35,39,42,24,0,0,0,0,0,108,0,0,0,0,0,0,1,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108],[41,75,0,0,0,0,0,0,9,67,0,0,0,0,0,0,0,0,35,74,73,6,71,103,0,0,35,74,73,6,44,71,0,0,108,107,0,105,1,18,0,0,1,108,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[68,5,0,0,0,0,0,0,100,41,0,0,0,0,0,0,0,0,67,24,6,41,6,73,0,0,67,24,65,14,6,73,0,0,104,86,0,108,82,59,0,0,86,91,0,18,50,28,0,0,0,0,0,0,50,32,0,0,0,0,0,0,82,59] >;`

C33.5D9 in GAP, Magma, Sage, TeX

`C_3^3._5D_9`
`% in TeX`

`G:=Group("C3^3.5D9");`
`// GroupNames label`

`G:=SmallGroup(486,162);`
`// by ID`

`G=gap.SmallGroup(486,162);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,1520,824,867,8104,208,11669]);`
`// Polycyclic`

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

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