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## G = S3×C9⋊S3order 324 = 22·34

### Direct product of S3 and C9⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C9⋊S3
G = < a,b,c,d,e | a3=b2=c9=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1216 in 130 conjugacy classes, 34 normal (16 characteristic)
C1, C2 [×3], C3 [×2], C3 [×3], C3 [×4], C22, S3, S3 [×13], C6 [×5], C9 [×3], C9 [×3], C32 [×2], C32 [×3], C32 [×4], D6 [×5], D9 [×9], C18 [×3], C3×S3, C3×S3 [×3], C3×S3 [×4], C3⋊S3 [×10], C3×C6, C3×C9, C3×C9 [×3], C3×C9 [×4], C33, D18 [×3], S32 [×4], C2×C3⋊S3, C3×D9 [×3], S3×C9 [×3], C9⋊S3, C9⋊S3 [×8], C3×C18, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C9, S3×D9 [×3], C2×C9⋊S3, S3×C3⋊S3, S3×C3×C9, C3×C9⋊S3, C324D9, S3×C9⋊S3
Quotients: C1, C2 [×3], C22, S3 [×5], D6 [×5], D9 [×3], C3⋊S3, D18 [×3], S32 [×4], C2×C3⋊S3, C9⋊S3, S3×D9 [×3], C2×C9⋊S3, S3×C3⋊S3, S3×C9⋊S3

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 6A 6B 6C 6D 6E 9A ··· 9I 9J ··· 9R 18A ··· 18I order 1 2 2 2 3 ··· 3 3 3 3 3 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 27 81 2 ··· 2 4 4 4 4 6 6 6 6 54 2 ··· 2 4 ··· 4 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 S3 D6 D6 D9 D18 S32 S32 S3×D9 kernel S3×C9⋊S3 S3×C3×C9 C3×C9⋊S3 C32⋊4D9 S3×C9 C9⋊S3 S3×C32 C3×C9 C33 C3×S3 C32 C9 C32 C3 # reps 1 1 1 1 3 1 1 4 1 9 9 3 1 9

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 17 7 0 0 0 0 12 5 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 1 18 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[17,12,0,0,0,0,7,5,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,18,18,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×C9⋊S3 in GAP, Magma, Sage, TeX

S_3\times C_9\rtimes S_3
% in TeX

G:=Group("S3xC9:S3");
// GroupNames label

G:=SmallGroup(324,120);
// by ID

G=gap.SmallGroup(324,120);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1052,986,297,2164,3899]);
// Polycyclic

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