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## G = C32×C9⋊S3order 486 = 2·35

### Direct product of C32 and C9⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C9⋊S3
G = < a,b,c,d,e | a3=b3=c9=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: 852 in 288 conjugacy classes, 66 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C33, C33, C3×D9, C9⋊S3, S3×C32, C3×C3⋊S3, C32×C9, C32×C9, C34, C32×D9, C3×C9⋊S3, C32×C3⋊S3, C33×C9, C32×C9⋊S3
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×D9, C9⋊S3, S3×C32, C3×C3⋊S3, C32×D9, C3×C9⋊S3, C32×C3⋊S3, C32×C9⋊S3

Smallest permutation representation of C32×C9⋊S3
On 54 points
Generators in S54
(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 40 29)(2 41 30)(3 42 31)(4 43 32)(5 44 33)(6 45 34)(7 37 35)(8 38 36)(9 39 28)(10 24 51)(11 25 52)(12 26 53)(13 27 54)(14 19 46)(15 20 47)(16 21 48)(17 22 49)(18 23 50)
(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 40 29)(2 41 30)(3 42 31)(4 43 32)(5 44 33)(6 45 34)(7 37 35)(8 38 36)(9 39 28)(10 51 24)(11 52 25)(12 53 26)(13 54 27)(14 46 19)(15 47 20)(16 48 21)(17 49 22)(18 50 23)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 54)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 45)(17 44)(18 43)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)

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

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

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

135 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AR 6A ··· 6H 9A ··· 9CC order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 1 ··· 1 2 ··· 2 27 ··· 27 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 C3×D9 kernel C32×C9⋊S3 C33×C9 C3×C9⋊S3 C32×C9 C32×C9 C34 C3×C9 C33 C33 C32 # reps 1 1 8 8 3 1 24 9 8 72

Matrix representation of C32×C9⋊S3 in GL4(𝔽19) generated by

 11 0 0 0 0 11 0 0 0 0 7 0 0 0 0 7
,
 11 0 0 0 0 11 0 0 0 0 1 0 0 0 0 1
,
 5 0 0 0 0 4 0 0 0 0 6 0 0 0 0 16
,
 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],[11,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,0,4,0,0,0,0,6,0,0,0,0,16],[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] >;

C32×C9⋊S3 in GAP, Magma, Sage, TeX

C_3^2\times C_9\rtimes S_3
% in TeX

G:=Group("C3^2xC9:S3");
// GroupNames label

G:=SmallGroup(486,227);
// by ID

G=gap.SmallGroup(486,227);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,453,3244,11669]);
// Polycyclic

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