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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, C3, C3, C3, C22, S3, S3, C6, C9, C9, C32, C32, C32, D6, D9, C18, C3×S3, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, D18, S32, C2×C3⋊S3, C3×D9, S3×C9, C9⋊S3, C9⋊S3, C3×C18, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C9, S3×D9, C2×C9⋊S3, S3×C3⋊S3, S3×C3×C9, C3×C9⋊S3, C324D9, S3×C9⋊S3
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, S32, C2×C3⋊S3, C9⋊S3, S3×D9, C2×C9⋊S3, S3×C3⋊S3, S3×C9⋊S3

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

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

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

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

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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