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## G = C32.C33order 243 = 35

### 9th non-split extension by C32 of C33 acting via C33/C32=C3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C32.16He3, C32.9C33, C33.15C32, 3- 1+2.3C32, C3.15(C3×He3), C3.He34C3, (C3×C9).14C32, (C3×3- 1+2).9C3, SmallGroup(243,59)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C32.C33
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — C32.C33
 Lower central C1 — C3 — C32 — C32.C33
 Upper central C1 — C3 — C33 — C32.C33
 Jennings C1 — C3 — C32 — C32.C33

Generators and relations for C32.C33
G = < a,b,c,d,e | a3=b3=e3=1, c3=b, d3=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ce=ec >

Subgroups: 126 in 62 conjugacy classes, 33 normal (6 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3.He3, C3×3- 1+2, C3×3- 1+2, C32.C33
Quotients: C1, C3, C32, He3, C33, C3×He3, C32.C33

Permutation representations of C32.C33
On 27 points - transitive group 27T111
Generators in S27
```(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)
(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)
(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)
(1 26 15 7 23 12 4 20 18)(2 21 13 8 27 10 5 24 16)(3 19 17 9 25 14 6 22 11)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)```

`G:=sub<Sym(27)| (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (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), (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), (1,26,15,7,23,12,4,20,18)(2,21,13,8,27,10,5,24,16)(3,19,17,9,25,14,6,22,11), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)>;`

`G:=Group( (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24), (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), (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), (1,26,15,7,23,12,4,20,18)(2,21,13,8,27,10,5,24,16)(3,19,17,9,25,14,6,22,11), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24) );`

`G=PermutationGroup([[(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,22,25),(21,27,24)], [(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)], [(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)], [(1,26,15,7,23,12,4,20,18),(2,21,13,8,27,10,5,24,16),(3,19,17,9,25,14,6,22,11)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)]])`

`G:=TransitiveGroup(27,111);`

C32.C33 is a maximal subgroup of   C3.He3⋊C6

35 conjugacy classes

 class 1 3A 3B 3C ··· 3J 9A ··· 9X order 1 3 3 3 ··· 3 9 ··· 9 size 1 1 1 3 ··· 3 9 ··· 9

35 irreducible representations

 dim 1 1 1 3 9 type + image C1 C3 C3 He3 C32.C33 kernel C32.C33 C3.He3 C3×3- 1+2 C32 C1 # reps 1 18 8 6 2

Matrix representation of C32.C33 in GL9(𝔽19)

 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 8 11 12 0 7 0 11 0 11 8 0 0 12 7 0 0 11
,
 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7
,
 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 18 18 0 1 1 6 0 18 1 0 0 18 1 1 0 6 7 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 18 0 1 0 0 0 0 1 18
,
 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 12 12 0 7 7 4 0 7 0 1 12 18 0 0 12 7 0 18 1 12 0 18 18 12 0
,
 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 7 18 0 12 1 7 0 11 7 0 0 12 0 8 0 11

`G:=sub<GL(9,GF(19))| [1,0,0,0,0,0,0,0,11,0,1,0,0,0,0,0,8,8,0,0,1,0,0,0,0,11,0,0,0,0,7,0,0,0,12,0,0,0,0,0,7,0,0,0,12,0,0,0,0,0,7,0,7,7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[0,0,0,0,0,18,7,18,18,0,0,0,0,1,1,0,1,0,0,0,0,0,18,0,0,0,1,1,0,0,0,18,0,0,0,0,0,1,0,0,0,18,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,6,0,0,0,1,0,0,0,0,0,6,0,18,18],[0,0,11,0,0,0,0,7,0,1,0,0,0,0,0,7,0,18,0,1,0,0,0,0,12,1,1,0,0,0,0,0,11,12,12,12,0,0,0,1,0,0,0,18,0,0,0,0,0,1,0,7,0,18,0,0,0,0,0,0,7,0,18,0,0,0,0,0,0,4,12,12,0,0,0,0,0,0,0,7,0],[1,0,0,0,0,0,0,0,11,0,7,0,0,0,0,0,0,7,0,0,11,0,0,0,0,7,0,0,0,0,1,0,0,0,18,0,0,0,0,0,7,0,0,0,12,0,0,0,0,0,11,0,12,0,0,0,0,0,0,0,1,1,8,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11] >;`

C32.C33 in GAP, Magma, Sage, TeX

`C_3^2.C_3^3`
`% in TeX`

`G:=Group("C3^2.C3^3");`
`// GroupNames label`

`G:=SmallGroup(243,59);`
`// by ID`

`G=gap.SmallGroup(243,59);`
`# by ID`

`G:=PCGroup([5,-3,3,3,-3,-3,405,301,546,457,2163]);`
`// Polycyclic`

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

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