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## G = C33⋊C4⋊C4order 432 = 24·33

### 2nd semidirect product of C33⋊C4 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C33⋊C4⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C33⋊9(C2×C4) — C33⋊C4⋊C4
 Lower central C33 — C3×C3⋊S3 — C33⋊C4⋊C4
 Upper central C1 — C2

Generators and relations for C33⋊C4⋊C4
G = < a,b,c,d,e | a3=b3=c3=d4=e4=1, ab=ba, ac=ca, dad-1=ab-1, ae=ea, bc=cb, dbd-1=ebe-1=a-1b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 580 in 96 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×4], C22, S3 [×4], C6, C6 [×6], C2×C4 [×3], C32, C32 [×4], Dic3 [×6], C12 [×4], D6 [×2], C2×C6, C4⋊C4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×6], C3⋊Dic3, C3×C12, C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, Dic3⋊C4, C3×C3⋊S3 [×2], C32×C6, S3×Dic3, C6.D6, C6.D6, S3×C12, C2×C32⋊C4, C32×Dic3, C3×C3⋊Dic3, C33⋊C4 [×2], C6×C3⋊S3, C3⋊S3.Q8, C3×C6.D6, C339(C2×C4), C2×C33⋊C4, C33⋊C4⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, Dic3⋊C4, S3≀C2, C3⋊S3.Q8, C33⋊D4, C33⋊C4⋊C4

Smallest permutation representation of C33⋊C4⋊C4
On 48 points
Generators in S48
```(1 16 45)(2 13 46)(3 47 14)(4 48 15)(5 28 38)(6 39 25)(7 40 26)(8 27 37)(9 17 34)(10 18 35)(11 36 19)(12 33 20)(21 29 43)(22 44 30)(23 41 31)(24 32 42)
(2 46 13)(4 15 48)(5 38 28)(7 26 40)(10 35 18)(12 20 33)(21 43 29)(23 31 41)
(1 45 16)(2 13 46)(3 47 14)(4 15 48)(5 38 28)(6 25 39)(7 40 26)(8 27 37)(9 34 17)(10 18 35)(11 36 19)(12 20 33)(21 29 43)(22 44 30)(23 31 41)(24 42 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)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 33 25 23)(2 36 26 22)(3 35 27 21)(4 34 28 24)(5 42 15 17)(6 41 16 20)(7 44 13 19)(8 43 14 18)(9 38 32 48)(10 37 29 47)(11 40 30 46)(12 39 31 45)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E ··· 12J 12K 12L order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 size 1 1 9 9 2 4 4 4 4 8 6 6 18 18 54 54 2 4 4 4 4 8 18 18 6 6 6 6 12 ··· 12 36 36

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 8 type + + + + + - + + - + + - image C1 C2 C2 C2 C4 S3 Q8 D4 D6 Dic6 C4×S3 C3⋊D4 S3≀C2 C3⋊S3.Q8 C33⋊D4 C33⋊C4⋊C4 C33⋊D4 C33⋊C4⋊C4 kernel C33⋊C4⋊C4 C3×C6.D6 C33⋊9(C2×C4) C2×C33⋊C4 C33⋊C4 C6.D6 C3×C3⋊S3 C32×C6 C2×C3⋊S3 C3⋊S3 C3⋊S3 C3×C6 C6 C3 C2 C1 C2 C1 # reps 1 1 1 1 4 1 1 1 1 2 2 2 4 4 4 4 1 1

Matrix representation of C33⋊C4⋊C4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 12 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 12 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 12 0 0 0 0 1 0 0 0 0 1 0 0
,
 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 2 0 0 0 0 11 4 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 1 0 1 0
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 8 0 8 0 0 8 0 8 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,1,0,0,0,0,12,0,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,0,12,0,1,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,5,0,8,0,0,5,0,8,0,0,0,0,0,0,8,0,0,0,0,8,0] >;`

C33⋊C4⋊C4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes C_4\rtimes C_4`
`% in TeX`

`G:=Group("C3^3:C4:C4");`
`// GroupNames label`

`G:=SmallGroup(432,581);`
`// by ID`

`G=gap.SmallGroup(432,581);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,1684,571,298,677,1027,14118]);`
`// Polycyclic`

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

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