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## G = C3⋊S3×C3×C9order 486 = 2·35

### Direct product of C3×C9 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C3×C9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C33×C9 — C3⋊S3×C3×C9
 Lower central C32 — C3⋊S3×C3×C9
 Upper central C1 — C3×C9

Generators and relations for C3⋊S3×C3×C9
G = < a,b,c,d,e | a3=b9=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 616 in 288 conjugacy classes, 70 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, C3×C9, C33, C33, C33, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C9, C32×C9, C34, S3×C3×C9, C9×C3⋊S3, C32×C3⋊S3, C33×C9, C3⋊S3×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, S3×C3×C9, C9×C3⋊S3, C32×C3⋊S3, C3⋊S3×C3×C9

Smallest permutation representation of C3⋊S3×C3×C9
On 54 points
Generators in S54
(1 36 42)(2 28 43)(3 29 44)(4 30 45)(5 31 37)(6 32 38)(7 33 39)(8 34 40)(9 35 41)(10 25 49)(11 26 50)(12 27 51)(13 19 52)(14 20 53)(15 21 54)(16 22 46)(17 23 47)(18 24 48)
(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 36 42)(2 28 43)(3 29 44)(4 30 45)(5 31 37)(6 32 38)(7 33 39)(8 34 40)(9 35 41)(10 49 25)(11 50 26)(12 51 27)(13 52 19)(14 53 20)(15 54 21)(16 46 22)(17 47 23)(18 48 24)
(1 45 33)(2 37 34)(3 38 35)(4 39 36)(5 40 28)(6 41 29)(7 42 30)(8 43 31)(9 44 32)(10 22 52)(11 23 53)(12 24 54)(13 25 46)(14 26 47)(15 27 48)(16 19 49)(17 20 50)(18 21 51)
(1 54)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 28)(17 29)(18 30)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 37)(26 38)(27 39)

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

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

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

162 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AR 6A ··· 6H 9A ··· 9R 9S ··· 9CL 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 kernel C3⋊S3×C3×C9 C33×C9 C9×C3⋊S3 C32×C3⋊S3 C32×C9 C34 C3×C3⋊S3 C33 C32×C9 C3×C9 C33 C32 # reps 1 1 6 2 6 2 18 18 4 24 8 72

Matrix representation of C3⋊S3×C3×C9 in GL4(𝔽19) generated by

 11 0 0 0 0 11 0 0 0 0 7 0 0 0 0 7
,
 6 0 0 0 0 6 0 0 0 0 4 0 0 0 0 4
,
 11 0 0 0 0 7 0 0 0 0 7 0 0 0 0 11
,
 11 0 0 0 0 7 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,7,0,0,0,0,7],[6,0,0,0,0,6,0,0,0,0,4,0,0,0,0,4],[11,0,0,0,0,7,0,0,0,0,7,0,0,0,0,11],[11,0,0,0,0,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C3×C9 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_3\times C_9
% in TeX

G:=Group("C3:S3xC3xC9");
// GroupNames label

G:=SmallGroup(486,228);
// by ID

G=gap.SmallGroup(486,228);
# by ID

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

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

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