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## G = (C32×C9)⋊S3order 486 = 2·35

### 14th semidirect product of C32×C9 and S3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — (C32×C9)⋊S3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C32.23C33 — (C32×C9)⋊S3
 Lower central C32⋊C9 — (C32×C9)⋊S3
 Upper central C1 — C3

Generators and relations for (C32×C9)⋊S3
G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, ac=ca, dad-1=ac3, ae=ea, bc=cb, dbd-1=bc3, be=eb, dcd-1=ab-1c, ece=c-1, ede=d-1 >

Subgroups: 524 in 99 conjugacy classes, 20 normal (13 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C322D9, C32×D9, C3×C32⋊C6, C3×C9⋊C6, C32.23C33, (C32×C9)⋊S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, He3⋊C2, C3×C3⋊S3, C3×He3⋊C2, He3.4S3, (C32×C9)⋊S3

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 6 6 type + + + + + + image C1 C2 C3 C6 S3 S3 S3 C3×S3 C3×S3 He3⋊C2 He3.4S3 (C32×C9)⋊S3 kernel (C32×C9)⋊S3 C32.23C33 C32⋊2D9 C32⋊C9 C32×C9 C3×He3 C3×3- 1+2 C3×C9 C33 C32 C3 C1 # reps 1 1 2 2 1 1 2 6 2 12 3 6

Matrix representation of (C32×C9)⋊S3 in GL6(𝔽19)

 7 15 0 0 0 0 12 12 1 0 0 0 11 18 0 0 0 0 0 0 0 7 15 0 0 0 0 12 12 1 0 0 0 11 18 0
,
 11 10 0 0 0 0 8 8 7 0 0 0 1 12 0 0 0 0 0 0 0 11 10 0 0 0 0 8 8 7 0 0 0 1 12 0
,
 6 0 14 0 0 0 9 0 15 0 0 0 10 6 13 0 0 0 0 0 0 17 12 0 0 0 0 2 2 16 0 0 0 5 3 0
,
 7 0 10 0 0 0 0 0 8 0 0 0 0 1 12 0 0 0 0 0 0 11 10 0 0 0 0 0 8 1 0 0 0 0 12 0
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(19))| [7,12,11,0,0,0,15,12,18,0,0,0,0,1,0,0,0,0,0,0,0,7,12,11,0,0,0,15,12,18,0,0,0,0,1,0],[11,8,1,0,0,0,10,8,12,0,0,0,0,7,0,0,0,0,0,0,0,11,8,1,0,0,0,10,8,12,0,0,0,0,7,0],[6,9,10,0,0,0,0,0,6,0,0,0,14,15,13,0,0,0,0,0,0,17,2,5,0,0,0,12,2,3,0,0,0,0,16,0],[7,0,0,0,0,0,0,0,1,0,0,0,10,8,12,0,0,0,0,0,0,11,0,0,0,0,0,10,8,12,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

(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(486,149);
// by ID

G=gap.SmallGroup(486,149);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,867,873,735,3244]);
// 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,d*a*d^-1=a*c^3,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^3,b*e=e*b,d*c*d^-1=a*b^-1*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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