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G = C62.9C32order 324 = 22·34

9th non-split extension by C62 of C32 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.9C32
 Chief series C1 — C22 — C2×C6 — C3×A4 — C9×A4 — C62.9C32
 Lower central C22 — C2×C6 — C62.9C32
 Upper central C1 — C3 — 3- 1+2

Generators and relations for C62.9C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab-1, dad-1=ab2, cbc-1=a3b4, bd=db, cd=dc >

Subgroups: 205 in 74 conjugacy classes, 36 normal (13 characteristic)
C1, C2, C3, C3, C22, C6, C9, C9, C32, C32, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, He3, 3- 1+2, 3- 1+2, C3.A4, C3.A4, C2×C18, C3×A4, C62, C2×3- 1+2, C9○He3, C9×A4, C9⋊A4, C3×C3.A4, C32.A4, C32⋊A4, C22×3- 1+2, C62.9C32
Quotients: C1, C3, C32, A4, C33, C3×A4, C9○He3, C32×A4, C62.9C32

Smallest permutation representation of C62.9C32
On 54 points
Generators in S54
```(2 5 8)(3 9 6)(10 46 16 52 13 49)(11 50 14 53 17 47)(12 54)(15 48)(18 51)(19 25 22)(21 24 27)(28 44 31 38 34 41)(29 39)(30 43 36 40 33 37)(32 42)(35 45)
(1 20 4 23 7 26)(2 21 5 24 8 27)(3 22 6 25 9 19)(10 16 13)(11 17 14)(12 18 15)(28 44 31 38 34 41)(29 45 32 39 35 42)(30 37 33 40 36 43)(46 52 49)(47 53 50)(48 54 51)
(1 49 31)(2 50 32)(3 51 33)(4 52 34)(5 53 35)(6 54 36)(7 46 28)(8 47 29)(9 48 30)(10 44 26)(11 45 27)(12 37 19)(13 38 20)(14 39 21)(15 40 22)(16 41 23)(17 42 24)(18 43 25)
(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)```

`G:=sub<Sym(54)| (2,5,8)(3,9,6)(10,46,16,52,13,49)(11,50,14,53,17,47)(12,54)(15,48)(18,51)(19,25,22)(21,24,27)(28,44,31,38,34,41)(29,39)(30,43,36,40,33,37)(32,42)(35,45), (1,20,4,23,7,26)(2,21,5,24,8,27)(3,22,6,25,9,19)(10,16,13)(11,17,14)(12,18,15)(28,44,31,38,34,41)(29,45,32,39,35,42)(30,37,33,40,36,43)(46,52,49)(47,53,50)(48,54,51), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,44,26)(11,45,27)(12,37,19)(13,38,20)(14,39,21)(15,40,22)(16,41,23)(17,42,24)(18,43,25), (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)>;`

`G:=Group( (2,5,8)(3,9,6)(10,46,16,52,13,49)(11,50,14,53,17,47)(12,54)(15,48)(18,51)(19,25,22)(21,24,27)(28,44,31,38,34,41)(29,39)(30,43,36,40,33,37)(32,42)(35,45), (1,20,4,23,7,26)(2,21,5,24,8,27)(3,22,6,25,9,19)(10,16,13)(11,17,14)(12,18,15)(28,44,31,38,34,41)(29,45,32,39,35,42)(30,37,33,40,36,43)(46,52,49)(47,53,50)(48,54,51), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,44,26)(11,45,27)(12,37,19)(13,38,20)(14,39,21)(15,40,22)(16,41,23)(17,42,24)(18,43,25), (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) );`

`G=PermutationGroup([[(2,5,8),(3,9,6),(10,46,16,52,13,49),(11,50,14,53,17,47),(12,54),(15,48),(18,51),(19,25,22),(21,24,27),(28,44,31,38,34,41),(29,39),(30,43,36,40,33,37),(32,42),(35,45)], [(1,20,4,23,7,26),(2,21,5,24,8,27),(3,22,6,25,9,19),(10,16,13),(11,17,14),(12,18,15),(28,44,31,38,34,41),(29,45,32,39,35,42),(30,37,33,40,36,43),(46,52,49),(47,53,50),(48,54,51)], [(1,49,31),(2,50,32),(3,51,33),(4,52,34),(5,53,35),(6,54,36),(7,46,28),(8,47,29),(9,48,30),(10,44,26),(11,45,27),(12,37,19),(13,38,20),(14,39,21),(15,40,22),(16,41,23),(17,42,24),(18,43,25)], [(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)]])`

44 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 6A 6B 6C 6D 9A ··· 9F 9G ··· 9L 9M ··· 9V 18A ··· 18F order 1 2 3 3 3 3 3 ··· 3 6 6 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 1 3 3 12 ··· 12 3 3 9 9 3 ··· 3 4 ··· 4 12 ··· 12 9 ··· 9

44 irreducible representations

 dim 1 1 1 1 1 1 1 3 3 3 3 9 type + + image C1 C3 C3 C3 C3 C3 C3 A4 C3×A4 C3×A4 C9○He3 C62.9C32 kernel C62.9C32 C9×A4 C9⋊A4 C3×C3.A4 C32.A4 C32⋊A4 C22×3- 1+2 3- 1+2 C9 C32 C22 C1 # reps 1 6 12 2 2 2 2 1 6 2 6 2

Matrix representation of C62.9C32 in GL6(𝔽19)

 1 0 0 0 0 0 0 18 0 0 0 0 3 0 18 0 0 0 0 0 0 7 1 12 0 0 0 0 11 0 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 18 0 0 0 0 16 16 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 1 0 0 0 0 3 3 17 0 0 0 4 6 16 0 0 0 0 0 0 9 4 6 0 0 0 0 0 9 0 0 0 5 13 10
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 11 7 0 0 0 0 0 1 0 0 0 9 12 18

`G:=sub<GL(6,GF(19))| [1,0,3,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,7,0,0,0,0,0,1,11,0,0,0,0,12,0,1],[18,0,16,0,0,0,0,18,16,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,3,4,0,0,0,1,3,6,0,0,0,0,17,16,0,0,0,0,0,0,9,0,5,0,0,0,4,0,13,0,0,0,6,9,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,9,0,0,0,11,0,12,0,0,0,7,1,18] >;`

C62.9C32 in GAP, Magma, Sage, TeX

`C_6^2._9C_3^2`
`% in TeX`

`G:=Group("C6^2.9C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,132);`
`// by ID`

`G=gap.SmallGroup(324,132);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,115,650,224,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=c^3=1,d^3=b^2,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b^2,c*b*c^-1=a^3*b^4,b*d=d*b,c*d=d*c>;`
`// generators/relations`

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