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## G = (C32×C9)⋊8S3order 486 = 2·35

### 8th semidirect product of C32×C9 and S3 acting faithfully

Aliases: (C32×C9)⋊8S3, C32⋊C911S3, (C3×He3).10C6, C33.21(C3×S3), He35S3.3C3, C32.23C333C2, C3.12(He34S3), C32.10(C32⋊C6), C3.2(He3.4C6), (C3×C9).4(C3⋊S3), C32.40(C3×C3⋊S3), SmallGroup(486,150)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — (C32×C9)⋊8S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C32.23C33 — (C32×C9)⋊8S3
 Lower central C3×He3 — (C32×C9)⋊8S3
 Upper central C1 — C3

Generators and relations for (C32×C9)⋊8S3
G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, ece=ac=ca, dad-1=ac6, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=bc7, ede=d-1 >

Subgroups: 596 in 96 conjugacy classes, 20 normal (11 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C32⋊C18, C9×C3⋊S3, He35S3, C32.23C33, (C32×C9)⋊8S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He34S3, He3.4C6, (C32×C9)⋊8S3

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 3 6 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 He3.4C6 C32⋊C6 (C32×C9)⋊8S3 kernel (C32×C9)⋊8S3 C32.23C33 He3⋊5S3 C3×He3 C32⋊C9 C32×C9 C33 C3 C32 C1 # reps 1 1 2 2 3 1 8 12 3 6

Matrix representation of (C32×C9)⋊8S3 in GL6(𝔽19)

 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 18 1 6 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 7 12 0 11 0 7 0 12 0 0 11
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 9 0 10 9 0 16 8 8 3 0 0 10 16 11 11 0 9 10
,
 0 0 1 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 11 8 11 9 0 0 0 7 12 8 7 0 0 8 11 8 0
,
 0 0 0 1 0 0 0 1 18 1 6 0 1 0 18 1 0 6 1 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 1

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

(C32×C9)⋊8S3 in GAP, Magma, Sage, TeX

(C_3^2\times C_9)\rtimes_8S_3
% in TeX

G:=Group("(C3^2xC9):8S3");
// GroupNames label

G:=SmallGroup(486,150);
// by ID

G=gap.SmallGroup(486,150);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,548,867,735,3244]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^9=d^3=e^2=1,a*b=b*a,e*c*e=a*c=c*a,d*a*d^-1=a*c^6,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=b*c^7,e*d*e=d^-1>;
// generators/relations

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