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## G = C62.C9order 324 = 22·34

### 2nd non-split extension by C62 of C9 acting via C9/C3=C3

Aliases: C62.2C9, C9.A42C3, C9.5(C3×A4), (C3×C9).3A4, C9.(C3.A4), (C2×C18).3C9, (C6×C18).2C3, C222(C27⋊C3), C32.(C3.A4), (C2×C18).5C32, (C2×C6).6(C3×C9), C3.3(C3×C3.A4), SmallGroup(324,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.C9
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9.A4 — C62.C9
 Lower central C22 — C2×C6 — C62.C9
 Upper central C1 — C9 — C3×C9

Generators and relations for C62.C9
G = < a,b,c | a6=b6=1, c9=b2, ab=ba, cac-1=ab-1, cbc-1=a3b4 >

Smallest permutation representation of C62.C9
On 54 points
Generators in S54
```(2 42 20 33 11 51)(3 52 12 34 21 43)(5 45 23 36 14 54)(6 28 15 37 24 46)(8 48 26 39 17 30)(9 31 18 40 27 49)
(1 50 10 32 19 41)(2 20 11)(3 52 12 34 21 43)(4 53 13 35 22 44)(5 23 14)(6 28 15 37 24 46)(7 29 16 38 25 47)(8 26 17)(9 31 18 40 27 49)(30 48 39)(33 51 42)(36 54 45)
(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)| (2,42,20,33,11,51)(3,52,12,34,21,43)(5,45,23,36,14,54)(6,28,15,37,24,46)(8,48,26,39,17,30)(9,31,18,40,27,49), (1,50,10,32,19,41)(2,20,11)(3,52,12,34,21,43)(4,53,13,35,22,44)(5,23,14)(6,28,15,37,24,46)(7,29,16,38,25,47)(8,26,17)(9,31,18,40,27,49)(30,48,39)(33,51,42)(36,54,45), (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( (2,42,20,33,11,51)(3,52,12,34,21,43)(5,45,23,36,14,54)(6,28,15,37,24,46)(8,48,26,39,17,30)(9,31,18,40,27,49), (1,50,10,32,19,41)(2,20,11)(3,52,12,34,21,43)(4,53,13,35,22,44)(5,23,14)(6,28,15,37,24,46)(7,29,16,38,25,47)(8,26,17)(9,31,18,40,27,49)(30,48,39)(33,51,42)(36,54,45), (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([[(2,42,20,33,11,51),(3,52,12,34,21,43),(5,45,23,36,14,54),(6,28,15,37,24,46),(8,48,26,39,17,30),(9,31,18,40,27,49)], [(1,50,10,32,19,41),(2,20,11),(3,52,12,34,21,43),(4,53,13,35,22,44),(5,23,14),(6,28,15,37,24,46),(7,29,16,38,25,47),(8,26,17),(9,31,18,40,27,49),(30,48,39),(33,51,42),(36,54,45)], [(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)]])`

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 6A ··· 6H 9A ··· 9F 9G 9H 9I 9J 18A ··· 18R 27A ··· 27R order 1 2 3 3 3 3 6 ··· 6 9 ··· 9 9 9 9 9 18 ··· 18 27 ··· 27 size 1 3 1 1 3 3 3 ··· 3 1 ··· 1 3 3 3 3 3 ··· 3 12 ··· 12

60 irreducible representations

 dim 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C9 C9 A4 C3.A4 C3×A4 C3.A4 C27⋊C3 C62.C9 kernel C62.C9 C9.A4 C6×C18 C2×C18 C62 C3×C9 C9 C9 C32 C22 C1 # reps 1 6 2 12 6 1 4 2 2 6 18

Matrix representation of C62.C9 in GL3(𝔽109) generated by

 1 0 0 101 64 0 90 0 46
,
 64 0 0 0 64 0 71 0 45
,
 66 107 0 0 43 1 53 4 0
`G:=sub<GL(3,GF(109))| [1,101,90,0,64,0,0,0,46],[64,0,71,0,64,0,0,0,45],[66,0,53,107,43,4,0,1,0] >;`

C62.C9 in GAP, Magma, Sage, TeX

`C_6^2.C_9`
`% in TeX`

`G:=Group("C6^2.C9");`
`// GroupNames label`

`G:=SmallGroup(324,45);`
`// by ID`

`G=gap.SmallGroup(324,45);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,361,68,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^6=1,c^9=b^2,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^4>;`
`// generators/relations`

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