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## G = C33⋊4C16order 432 = 24·33

### 2nd semidirect product of C33 and C16 acting via C16/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊4C16
 Chief series C1 — C3 — C33 — C32×C6 — C32×C12 — C3×C32⋊4C8 — C33⋊4C16
 Lower central C33 — C33⋊4C16
 Upper central C1 — C4

Generators and relations for C334C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >

Smallest permutation representation of C334C16
On 48 points
Generators in S48
```(1 18 46)(2 19 47)(3 48 20)(4 33 21)(5 22 34)(6 23 35)(7 36 24)(8 37 25)(9 26 38)(10 27 39)(11 40 28)(12 41 29)(13 30 42)(14 31 43)(15 44 32)(16 45 17)
(2 47 19)(4 21 33)(6 35 23)(8 25 37)(10 39 27)(12 29 41)(14 43 31)(16 17 45)
(1 46 18)(2 19 47)(3 48 20)(4 21 33)(5 34 22)(6 23 35)(7 36 24)(8 25 37)(9 38 26)(10 27 39)(11 40 28)(12 29 41)(13 42 30)(14 31 43)(15 44 32)(16 17 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)```

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

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

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

48 conjugacy classes

 class 1 2 3A 3B ··· 3G 4A 4B 6A 6B ··· 6G 8A 8B 8C 8D 12A 12B 12C ··· 12N 16A ··· 16H 24A 24B 24C 24D order 1 2 3 3 ··· 3 4 4 6 6 ··· 6 8 8 8 8 12 12 12 ··· 12 16 ··· 16 24 24 24 24 size 1 1 2 4 ··· 4 1 1 2 4 ··· 4 9 9 9 9 2 2 4 ··· 4 27 ··· 27 18 18 18 18

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + - + - image C1 C2 C4 C8 C16 S3 Dic3 C3⋊C8 C3⋊C16 C32⋊C4 C32⋊2C8 C33⋊C4 C32⋊2C16 C33⋊4C8 C33⋊4C16 kernel C33⋊4C16 C3×C32⋊4C8 C32×C12 C32×C6 C33 C32⋊4C8 C3×C12 C3×C6 C32 C12 C6 C4 C3 C2 C1 # reps 1 1 2 4 8 1 1 2 4 2 2 4 4 4 8

Matrix representation of C334C16 in GL4(𝔽97) generated by

 35 0 0 0 0 61 0 0 0 0 61 0 0 0 0 35
,
 1 0 0 0 0 1 0 0 0 0 35 0 0 0 0 61
,
 61 0 0 0 0 61 0 0 0 0 35 0 0 0 0 35
,
 0 0 1 0 0 0 0 1 0 1 0 0 22 0 0 0
`G:=sub<GL(4,GF(97))| [35,0,0,0,0,61,0,0,0,0,61,0,0,0,0,35],[1,0,0,0,0,1,0,0,0,0,35,0,0,0,0,61],[61,0,0,0,0,61,0,0,0,0,35,0,0,0,0,35],[0,0,0,22,0,0,1,0,1,0,0,0,0,1,0,0] >;`

C334C16 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_4C_{16}`
`% in TeX`

`G:=Group("C3^3:4C16");`
`// GroupNames label`

`G:=SmallGroup(432,413);`
`// by ID`

`G=gap.SmallGroup(432,413);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,14,36,58,2804,571,2693,2028,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;`
`// generators/relations`

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