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## G = C3.C3≀S3order 486 = 2·35

### 1st non-split extension by C3 of C3≀S3 acting via C3≀S3/C3≀C3=C2

Aliases: C3.1C3≀S3, (C3×He3)⋊1S3, (C3×He3)⋊1C6, C33.1(C3×S3), He35S31C3, C3.3(C33⋊C6), C32.24He31C2, C32.25(C32⋊C6), SmallGroup(486,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — C3.C3≀S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C32.24He3 — C3.C3≀S3
 Lower central C3×He3 — C3.C3≀S3
 Upper central C1 — C3

Generators and relations for C3.C3≀S3
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, eae-1=a-1, bc=cb, dbd-1=bc-1, ebe-1=ab-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Subgroups: 704 in 73 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3 [×2], C3 [×8], S3 [×5], C6 [×4], C9, C32, C32 [×13], C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C9, He3 [×6], C33, C33 [×2], C32⋊C6 [×3], He3⋊C2 [×3], S3×C32, C3×C3⋊S3 [×2], C32⋊C9, C3×He3 [×2], C3×C32⋊C6, He35S3, C32.24He3, C3.C3≀S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C32⋊C6, C3≀S3, C33⋊C6, C3.C3≀S3

Smallest permutation representation of C3.C3≀S3
On 54 points
Generators in S54
```(1 17 24)(2 19 18)(3 13 20)(4 21 14)(5 15 22)(6 23 16)(7 25 53)(8 54 26)(9 27 49)(10 50 28)(11 29 51)(12 52 30)(31 46 37)(32 38 47)(33 48 39)(34 40 43)(35 44 41)(36 42 45)
(2 19 18)(3 13 20)(4 14 21)(5 22 15)(8 54 26)(9 27 49)(10 28 50)(11 51 29)(31 37 46)(32 47 38)(35 44 41)(36 42 45)
(1 17 24)(2 18 19)(3 13 20)(4 14 21)(5 15 22)(6 16 23)(7 25 53)(8 26 54)(9 27 49)(10 28 50)(11 29 51)(12 30 52)(31 37 46)(32 38 47)(33 39 48)(34 40 43)(35 41 44)(36 42 45)
(1 29 36)(2 31 12)(3 25 38)(4 33 26)(5 9 34)(6 41 28)(7 32 20)(8 21 48)(10 23 35)(11 45 24)(13 53 47)(14 39 54)(15 27 40)(16 44 50)(17 51 42)(18 37 30)(19 46 52)(22 49 43)
(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)| (1,17,24)(2,19,18)(3,13,20)(4,21,14)(5,15,22)(6,23,16)(7,25,53)(8,54,26)(9,27,49)(10,50,28)(11,29,51)(12,52,30)(31,46,37)(32,38,47)(33,48,39)(34,40,43)(35,44,41)(36,42,45), (2,19,18)(3,13,20)(4,14,21)(5,22,15)(8,54,26)(9,27,49)(10,28,50)(11,51,29)(31,37,46)(32,47,38)(35,44,41)(36,42,45), (1,17,24)(2,18,19)(3,13,20)(4,14,21)(5,15,22)(6,16,23)(7,25,53)(8,26,54)(9,27,49)(10,28,50)(11,29,51)(12,30,52)(31,37,46)(32,38,47)(33,39,48)(34,40,43)(35,41,44)(36,42,45), (1,29,36)(2,31,12)(3,25,38)(4,33,26)(5,9,34)(6,41,28)(7,32,20)(8,21,48)(10,23,35)(11,45,24)(13,53,47)(14,39,54)(15,27,40)(16,44,50)(17,51,42)(18,37,30)(19,46,52)(22,49,43), (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( (1,17,24)(2,19,18)(3,13,20)(4,21,14)(5,15,22)(6,23,16)(7,25,53)(8,54,26)(9,27,49)(10,50,28)(11,29,51)(12,52,30)(31,46,37)(32,38,47)(33,48,39)(34,40,43)(35,44,41)(36,42,45), (2,19,18)(3,13,20)(4,14,21)(5,22,15)(8,54,26)(9,27,49)(10,28,50)(11,51,29)(31,37,46)(32,47,38)(35,44,41)(36,42,45), (1,17,24)(2,18,19)(3,13,20)(4,14,21)(5,15,22)(6,16,23)(7,25,53)(8,26,54)(9,27,49)(10,28,50)(11,29,51)(12,30,52)(31,37,46)(32,38,47)(33,39,48)(34,40,43)(35,41,44)(36,42,45), (1,29,36)(2,31,12)(3,25,38)(4,33,26)(5,9,34)(6,41,28)(7,32,20)(8,21,48)(10,23,35)(11,45,24)(13,53,47)(14,39,54)(15,27,40)(16,44,50)(17,51,42)(18,37,30)(19,46,52)(22,49,43), (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([(1,17,24),(2,19,18),(3,13,20),(4,21,14),(5,15,22),(6,23,16),(7,25,53),(8,54,26),(9,27,49),(10,50,28),(11,29,51),(12,52,30),(31,46,37),(32,38,47),(33,48,39),(34,40,43),(35,44,41),(36,42,45)], [(2,19,18),(3,13,20),(4,14,21),(5,22,15),(8,54,26),(9,27,49),(10,28,50),(11,51,29),(31,37,46),(32,47,38),(35,44,41),(36,42,45)], [(1,17,24),(2,18,19),(3,13,20),(4,14,21),(5,15,22),(6,16,23),(7,25,53),(8,26,54),(9,27,49),(10,28,50),(11,29,51),(12,30,52),(31,37,46),(32,38,47),(33,39,48),(34,40,43),(35,41,44),(36,42,45)], [(1,29,36),(2,31,12),(3,25,38),(4,33,26),(5,9,34),(6,41,28),(7,32,20),(8,21,48),(10,23,35),(11,45,24),(13,53,47),(14,39,54),(15,27,40),(16,44,50),(17,51,42),(18,37,30),(19,46,52),(22,49,43)], [(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)])`

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L 3M 3N 3O 6A ··· 6H 9A ··· 9F order 1 2 3 3 3 3 3 3 ··· 3 3 3 3 3 6 ··· 6 9 ··· 9 size 1 27 1 1 2 2 2 9 ··· 9 18 18 18 18 27 ··· 27 18 ··· 18

31 irreducible representations

 dim 1 1 1 1 2 2 3 6 6 6 type + + + + + image C1 C2 C3 C6 S3 C3×S3 C3≀S3 C32⋊C6 C33⋊C6 C3.C3≀S3 kernel C3.C3≀S3 C32.24He3 He3⋊5S3 C3×He3 C3×He3 C33 C3 C32 C3 C1 # reps 1 1 2 2 1 2 12 1 3 6

Matrix representation of C3.C3≀S3 in GL6(𝔽19)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 0 0 0 0 0 17 0 0 0 17 0 0 0 0 0 0 5 0 0 0 17 0 0 0 17 0 0 0 0 0 0 5 0 0 0 0

`G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,0,0,17,0,0,0,0,0,0,5,0,0,0,17,0,0,0,17,0,0,0,0,0,0,5,0,0,0,17,0,0,0,0,0] >;`

C3.C3≀S3 in GAP, Magma, Sage, TeX

`C_3.C_3\wr S_3`
`% in TeX`

`G:=Group("C3.C3wrS3");`
`// GroupNames label`

`G:=SmallGroup(486,4);`
`// by ID`

`G=gap.SmallGroup(486,4);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,224,4755,873,735,3244]);`
`// Polycyclic`

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

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