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## G = C3⋊S3×3- 1+2order 486 = 2·35

### Direct product of C3⋊S3 and 3- 1+2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3⋊S3×3- 1+2
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32×3- 1+2 — C3⋊S3×3- 1+2
 Lower central C32 — C33 — C3⋊S3×3- 1+2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for C3⋊S3×3- 1+2
G = < a,b,c,d,e | a3=b3=c2=d9=e3=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 616 in 216 conjugacy classes, 49 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, S3×C9, C2×3- 1+2, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C9, C3×3- 1+2, C3×3- 1+2, C34, S3×3- 1+2, C9×C3⋊S3, C32×C3⋊S3, C32×3- 1+2, C3⋊S3×3- 1+2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, 3- 1+2, C2×3- 1+2, S3×C32, C3×C3⋊S3, S3×3- 1+2, C32×C3⋊S3, C3⋊S3×3- 1+2

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

G:=sub<Sym(54)| (1,38,28)(2,39,29)(3,40,30)(4,41,31)(5,42,32)(6,43,33)(7,44,34)(8,45,35)(9,37,36)(10,23,50)(11,24,51)(12,25,52)(13,26,53)(14,27,54)(15,19,46)(16,20,47)(17,21,48)(18,22,49), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53)>;

G:=Group( (1,38,28)(2,39,29)(3,40,30)(4,41,31)(5,42,32)(6,43,33)(7,44,34)(8,45,35)(9,37,36)(10,23,50)(11,24,51)(12,25,52)(13,26,53)(14,27,54)(15,19,46)(16,20,47)(17,21,48)(18,22,49), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53) );

G=PermutationGroup([[(1,38,28),(2,39,29),(3,40,30),(4,41,31),(5,42,32),(6,43,33),(7,44,34),(8,45,35),(9,37,36),(10,23,50),(11,24,51),(12,25,52),(13,26,53),(14,27,54),(15,19,46),(16,20,47),(17,21,48),(18,22,49)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(19,45),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(37,40,43),(39,45,42),(46,52,49),(47,50,53)]])

66 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 3O 3P 3Q ··· 3X 6A 6B 6C 6D 9A ··· 9F 9G ··· 9AD 18A ··· 18F order 1 2 3 3 3 ··· 3 3 3 3 ··· 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 ··· 2 3 3 6 ··· 6 9 9 27 27 3 ··· 3 6 ··· 6 27 ··· 27

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 S3×3- 1+2 kernel C3⋊S3×3- 1+2 C32×3- 1+2 C9×C3⋊S3 C32×C3⋊S3 C32×C9 C34 C3×3- 1+2 C3×C9 C33 C3⋊S3 C32 C3 # reps 1 1 6 2 6 2 4 24 8 2 2 8

Matrix representation of C3⋊S3×3- 1+2 in GL7(𝔽19)

 0 18 0 0 0 0 0 1 18 0 0 0 0 0 0 0 18 1 0 0 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 18 1 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 18 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 18 0 0 0 0 0 0 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 18 9 12 0 0 0 0 7 1 0 0 0 0 0 1 0 0
,
 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 11 0 0 0 0 0 0 18 1 11 0 0 0 0 0 0 7

G:=sub<GL(7,GF(19))| [0,1,0,0,0,0,0,18,18,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,18,7,1,0,0,0,0,9,1,0,0,0,0,0,12,0,0],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11,18,0,0,0,0,0,0,1,0,0,0,0,0,0,11,7] >;

C3⋊S3×3- 1+2 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times 3_-^{1+2}
% in TeX

G:=Group("C3:S3xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,233);
// by ID

G=gap.SmallGroup(486,233);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^9=e^3=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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