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G = S3×C9⋊S3order 324 = 22·34

Direct product of S3 and C9⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×C9⋊S3, C324D18, C33.9D6, C91S32, (S3×C9)⋊S3, (C3×S3)⋊D9, C32(S3×D9), (C3×C9)⋊17D6, C32.11S32, (C32×C9)⋊4C22, C324D92C2, (S3×C32).3S3, C31(C2×C9⋊S3), (S3×C3×C9)⋊3C2, (C3×C9⋊S3)⋊2C2, C3.1(S3×C3⋊S3), (C3×S3).(C3⋊S3), C32.7(C2×C3⋊S3), SmallGroup(324,120)

Series: Derived Chief Lower central Upper central

C1C32×C9 — S3×C9⋊S3
C1C3C32C33C32×C9S3×C3×C9 — S3×C9⋊S3
C32×C9 — S3×C9⋊S3
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 2A2B2C3A···3E3F3G3H3I6A6B6C6D6E9A···9I9J···9R18A···18I
order12223···33333666669···99···918···18
size1327812···244446666542···24···46···6

45 irreducible representations

dim11112222222444
type++++++++++++++
imageC1C2C2C2S3S3S3D6D6D9D18S32S32S3×D9
kernelS3×C9⋊S3S3×C3×C9C3×C9⋊S3C324D9S3×C9C9⋊S3S3×C32C3×C9C33C3×S3C32C9C32C3
# reps11113114199319

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

100000
010000
001000
000100
0000181
0000180
,
1800000
0180000
0018000
0001800
000001
000010
,
1770000
1250000
0001800
0011800
000010
000001
,
0180000
1180000
0001800
0011800
000010
000001
,
0180000
1800000
0001800
0018000
000010
000001

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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