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## G = C62.13C32order 324 = 22·34

### 4th non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.13C32, (C3×C9)⋊1A4, (C6×C18)⋊1C3, (C2×C6).1He3, C32⋊A4.1C3, C32.A41C3, C3.3(C32⋊A4), C32.10(C3×A4), C221(He3.C3), SmallGroup(324,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.13C32
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — C62.13C32
 Lower central C22 — C2×C6 — C62 — C62.13C32
 Upper central C1 — C3 — C32 — C3×C9

Generators and relations for C62.13C32
G = < a,b,c,d | a6=b6=c3=1, d3=b2, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b4, bd=db, dcd-1=a2b4c >

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

44 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C3 A4 He3 C3×A4 He3.C3 C32⋊A4 C62.13C32 kernel C62.13C32 C32.A4 C32⋊A4 C6×C18 C3×C9 C2×C6 C32 C22 C3 C1 # reps 1 4 2 2 1 2 2 6 6 18

Matrix representation of C62.13C32 in GL3(𝔽19) generated by

 18 0 0 0 7 0 0 0 8
,
 7 0 0 0 12 0 0 0 12
,
 0 1 0 0 0 1 1 0 0
,
 16 0 0 0 16 0 0 0 5
`G:=sub<GL(3,GF(19))| [18,0,0,0,7,0,0,0,8],[7,0,0,0,12,0,0,0,12],[0,0,1,1,0,0,0,1,0],[16,0,0,0,16,0,0,0,5] >;`

C62.13C32 in GAP, Magma, Sage, TeX

`C_6^2._{13}C_3^2`
`% in TeX`

`G:=Group("C6^2.13C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,49);`
`// by ID`

`G=gap.SmallGroup(324,49);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,145,115,650,4864,8753]);`
`// 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,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=a^2*b^4*c>;`
`// generators/relations`

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