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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 — C32 — C3×S3×3- 1+2
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C32×3- 1+2 — C3×S3×3- 1+2
 Lower central C3 — C32 — C3×S3×3- 1+2
 Upper central C1 — C32 — 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, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 596 in 258 conjugacy classes, 96 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, S3×C9, C3×C18, C2×3- 1+2, S3×C32, S3×C32, S3×C32, C32×C6, C32×C9, C3×3- 1+2, C3×3- 1+2, C3×3- 1+2, C34, S3×C3×C9, S3×3- 1+2, C6×3- 1+2, S3×C33, C32×3- 1+2, C3×S3×3- 1+2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C33, C2×3- 1+2, S3×C32, C32×C6, C3×3- 1+2, S3×3- 1+2, C6×3- 1+2, S3×C33, C3×S3×3- 1+2

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

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

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

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

99 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3W 3X ··· 3AC 6A ··· 6H 6I ··· 6N 9A ··· 9R 9S ··· 9AJ 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 3 ··· 3 9 ··· 9 3 ··· 3 6 ··· 6 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×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 S3×C3×C9 S3×3- 1+2 S3×C33 C32×C9 C3×3- 1+2 C34 C3×3- 1+2 C3×C9 3- 1+2 C33 C3×S3 C32 C3 # reps 1 1 6 18 2 6 18 2 1 6 18 2 6 6 6

Matrix representation of C3×S3×3- 1+2 in GL5(𝔽19)

 11 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 0 0 0 0 0 11 0 0 0 0 0 11 11 11 0 0 0 0 7 0 0 15 8 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 8 7 0 0 0 11 0 0 0 0 0 1

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,15,0,0,11,0,8,0,0,11,7,8],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,8,11,0,0,0,7,0,1] >;

C3×S3×3- 1+2 in GAP, Magma, Sage, TeX

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

G:=Group("C3xS3xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,225);
// by ID

G=gap.SmallGroup(486,225);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,303,93,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^9=e^3=1,a*b=b*a,a*c=c*a,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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