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## G = C3.A42order 432 = 24·33

### 2nd central extension by C3 of A42

Aliases: C3.2A42, C22⋊A4⋊C9, C3.A43A4, C242(C3×C9), C221(C9×A4), C24⋊C92C3, (C23×C6).2C32, (C2×C6).2(C3×A4), (C3×C22⋊A4).1C3, (C22×C3.A4)⋊2C3, SmallGroup(432,525)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3.A42
 Chief series C1 — C22 — C24 — C23×C6 — C22×C3.A4 — C3.A42
 Lower central C24 — C3.A42
 Upper central C1 — C3

Generators and relations for C3.A42
G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f9=1, eae-1=fbf-1=ab=ba, fcf-1=ac=ca, ad=da, faf-1=b, bc=cb, fdf-1=bd=db, ebe-1=a, ece-1=cd=dc, ede-1=c, ef=fe >

Subgroups: 394 in 71 conjugacy classes, 18 normal (7 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C32, A4, C2×C6, C2×C6, C24, C18, C22×C6, C3×C9, C3.A4, C3.A4, C2×C18, C3×A4, C22⋊A4, C23×C6, C2×C3.A4, C9×A4, C22×C3.A4, C24⋊C9, C3×C22⋊A4, C3.A42
Quotients: C1, C3, C9, C32, A4, C3×C9, C3×A4, C9×A4, A42, C3.A42

Smallest permutation representation of C3.A42
On 36 points
Generators in S36
```(1 12)(2 24)(3 34)(4 15)(5 27)(6 28)(7 18)(8 21)(9 31)(10 30)(11 22)(13 33)(14 25)(16 36)(17 19)(20 29)(23 32)(26 35)
(1 23)(2 33)(3 14)(4 26)(5 36)(6 17)(7 20)(8 30)(9 11)(10 21)(12 32)(13 24)(15 35)(16 27)(18 29)(19 28)(22 31)(25 34)
(2 13)(3 34)(5 16)(6 28)(8 10)(9 31)(11 22)(14 25)(17 19)(21 30)(24 33)(27 36)
(2 24)(3 14)(5 27)(6 17)(8 21)(9 11)(10 30)(13 33)(16 36)(19 28)(22 31)(25 34)
(1 7 4)(2 8 5)(3 9 6)(10 27 33)(11 19 34)(12 20 35)(13 21 36)(14 22 28)(15 23 29)(16 24 30)(17 25 31)(18 26 32)
(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)```

`G:=sub<Sym(36)| (1,12)(2,24)(3,34)(4,15)(5,27)(6,28)(7,18)(8,21)(9,31)(10,30)(11,22)(13,33)(14,25)(16,36)(17,19)(20,29)(23,32)(26,35), (1,23)(2,33)(3,14)(4,26)(5,36)(6,17)(7,20)(8,30)(9,11)(10,21)(12,32)(13,24)(15,35)(16,27)(18,29)(19,28)(22,31)(25,34), (2,13)(3,34)(5,16)(6,28)(8,10)(9,31)(11,22)(14,25)(17,19)(21,30)(24,33)(27,36), (2,24)(3,14)(5,27)(6,17)(8,21)(9,11)(10,30)(13,33)(16,36)(19,28)(22,31)(25,34), (1,7,4)(2,8,5)(3,9,6)(10,27,33)(11,19,34)(12,20,35)(13,21,36)(14,22,28)(15,23,29)(16,24,30)(17,25,31)(18,26,32), (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)>;`

`G:=Group( (1,12)(2,24)(3,34)(4,15)(5,27)(6,28)(7,18)(8,21)(9,31)(10,30)(11,22)(13,33)(14,25)(16,36)(17,19)(20,29)(23,32)(26,35), (1,23)(2,33)(3,14)(4,26)(5,36)(6,17)(7,20)(8,30)(9,11)(10,21)(12,32)(13,24)(15,35)(16,27)(18,29)(19,28)(22,31)(25,34), (2,13)(3,34)(5,16)(6,28)(8,10)(9,31)(11,22)(14,25)(17,19)(21,30)(24,33)(27,36), (2,24)(3,14)(5,27)(6,17)(8,21)(9,11)(10,30)(13,33)(16,36)(19,28)(22,31)(25,34), (1,7,4)(2,8,5)(3,9,6)(10,27,33)(11,19,34)(12,20,35)(13,21,36)(14,22,28)(15,23,29)(16,24,30)(17,25,31)(18,26,32), (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) );`

`G=PermutationGroup([[(1,12),(2,24),(3,34),(4,15),(5,27),(6,28),(7,18),(8,21),(9,31),(10,30),(11,22),(13,33),(14,25),(16,36),(17,19),(20,29),(23,32),(26,35)], [(1,23),(2,33),(3,14),(4,26),(5,36),(6,17),(7,20),(8,30),(9,11),(10,21),(12,32),(13,24),(15,35),(16,27),(18,29),(19,28),(22,31),(25,34)], [(2,13),(3,34),(5,16),(6,28),(8,10),(9,31),(11,22),(14,25),(17,19),(21,30),(24,33),(27,36)], [(2,24),(3,14),(5,27),(6,17),(8,21),(9,11),(10,30),(13,33),(16,36),(19,28),(22,31),(25,34)], [(1,7,4),(2,8,5),(3,9,6),(10,27,33),(11,19,34),(12,20,35),(13,21,36),(14,22,28),(15,23,29),(16,24,30),(17,25,31),(18,26,32)], [(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)]])`

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 6A 6B 6C 6D 6E 6F 9A ··· 9L 9M ··· 9R 18A ··· 18L order 1 2 2 2 3 3 3 ··· 3 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 3 9 1 1 16 ··· 16 3 3 3 3 9 9 4 ··· 4 16 ··· 16 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 3 3 3 9 9 type + + + image C1 C3 C3 C3 C9 A4 C3×A4 C9×A4 A42 C3.A42 kernel C3.A42 C22×C3.A4 C24⋊C9 C3×C22⋊A4 C22⋊A4 C3.A4 C2×C6 C22 C3 C1 # reps 1 4 2 2 18 2 4 12 1 2

Matrix representation of C3.A42 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 18 18 18 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 18 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 18 18 18 0 0 0 0 0 0 18 18 18 0 0 0 0 0 1 0 0 0 0 1 0
,
 18 18 18 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 18 18 18
,
 7 0 0 0 0 0 0 0 7 0 0 0 12 12 12 0 0 0 0 0 0 1 0 0 0 0 0 18 18 18 0 0 0 0 1 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 18 18 18

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

C3.A42 in GAP, Magma, Sage, TeX

`C_3.A_4^2`
`% in TeX`

`G:=Group("C3.A4^2");`
`// GroupNames label`

`G:=SmallGroup(432,525);`
`// by ID`

`G=gap.SmallGroup(432,525);`
`# by ID`

`G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,50,766,326,13613,5298]);`
`// Polycyclic`

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

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