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## G = C33.31C32order 243 = 35

### 14th non-split extension by C33 of C32 acting via C32/C3=C3

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C33.31C32, C32.25C33, C32.83- 1+2, C9⋊C94C3, C32⋊C9.9C3, (C3×C9).8C32, (C32×C9).7C3, C3.7(C9○He3), C3.7(C3×3- 1+2), SmallGroup(243,42)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C33.31C32
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C33.31C32
 Lower central C1 — C32 — C33.31C32
 Upper central C1 — C32 — C33.31C32
 Jennings C1 — C32 — C32 — C33.31C32

Generators and relations for C33.31C32
G = < a,b,c,d,e | a3=b3=d3=1, c3=b, e3=a, ab=ba, dcd-1=ac=ca, ad=da, ae=ea, ece-1=bc=cb, bd=db, be=eb, de=ed >

Subgroups: 99 in 57 conjugacy classes, 36 normal (6 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, C33, C32⋊C9, C9⋊C9, C32×C9, C33.31C32
Quotients: C1, C3, C32, 3- 1+2, C33, C3×3- 1+2, C9○He3, C33.31C32

Smallest permutation representation of C33.31C32
On 81 points
Generators in S81
```(1 80 67)(2 81 68)(3 73 69)(4 74 70)(5 75 71)(6 76 72)(7 77 64)(8 78 65)(9 79 66)(10 53 56)(11 54 57)(12 46 58)(13 47 59)(14 48 60)(15 49 61)(16 50 62)(17 51 63)(18 52 55)(19 32 37)(20 33 38)(21 34 39)(22 35 40)(23 36 41)(24 28 42)(25 29 43)(26 30 44)(27 31 45)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 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 7 4)(2 65 75)(3 79 72)(5 68 78)(6 73 66)(8 71 81)(9 76 69)(10 16 13)(11 63 48)(12 52 61)(14 57 51)(15 46 55)(17 60 54)(18 49 58)(19 29 40)(20 26 23)(21 45 28)(22 32 43)(24 39 31)(25 35 37)(27 42 34)(30 36 33)(38 44 41)(47 53 50)(56 62 59)(64 70 67)(74 80 77)
(1 56 36 80 10 41 67 53 23)(2 63 31 81 17 45 68 51 27)(3 61 35 73 15 40 69 49 22)(4 59 30 74 13 44 70 47 26)(5 57 34 75 11 39 71 54 21)(6 55 29 76 18 43 72 52 25)(7 62 33 77 16 38 64 50 20)(8 60 28 78 14 42 65 48 24)(9 58 32 79 12 37 66 46 19)```

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

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

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

C33.31C32 is a maximal subgroup of   C9⋊C92S3

51 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 9A ··· 9R 9S ··· 9AJ order 1 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 size 1 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C3 C3 3- 1+2 C9○He3 kernel C33.31C32 C32⋊C9 C9⋊C9 C32×C9 C32 C3 # reps 1 6 18 2 6 18

Matrix representation of C33.31C32 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 6 0 0 0 0 0 18 1 0 0 0 1 18 0 0 0 0 0 0 0 1 11 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 11 0 0 0 0 0 12 7 0 0 0 0 11 0 1 0 0 0 0 0 0 1 0 0 0 0 0 13 11 0 0 0 0 4 0 7
,
 17 0 0 0 0 0 17 5 0 0 0 0 3 0 16 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5

`G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,1,0,0,0,6,18,18,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,18,18,0,0,0,0,1,0],[11,12,11,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,4,0,0,0,0,11,0,0,0,0,0,0,7],[17,17,3,0,0,0,0,5,0,0,0,0,0,0,16,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;`

C33.31C32 in GAP, Magma, Sage, TeX

`C_3^3._{31}C_3^2`
`% in TeX`

`G:=Group("C3^3.31C3^2");`
`// GroupNames label`

`G:=SmallGroup(243,42);`
`// by ID`

`G=gap.SmallGroup(243,42);`
`# by ID`

`G:=PCGroup([5,-3,3,3,-3,3,405,301,1352,57]);`
`// Polycyclic`

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

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