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## G = C9⋊(S3×C9)  order 486 = 2·35

### The semidirect product of C9 and S3×C9 acting via S3×C9/C32=C6

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9⋊(S3×C9)
G = < a,b,c,d | a9=b9=c3=d2=1, bab-1=a7, ac=ca, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 354 in 90 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C9⋊C9, C9⋊C9, C32×C9, C32×C9, C9⋊C18, C3×C9⋊S3, C9×C3⋊S3, C3×C9⋊C9, C9⋊(S3×C9)
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C9⋊C6, C3×C3⋊S3, C9⋊C18, C9×C3⋊S3, C33.S3, C9⋊(S3×C9)

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

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

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

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

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9F 9G ··· 9AM 18A ··· 18F order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 ··· 2 27 27 3 ··· 3 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + image C1 C2 C3 C6 C9 C18 S3 S3 C3×S3 C3×S3 S3×C9 S3×C9 C9⋊C6 C9⋊C18 kernel C9⋊(S3×C9) C3×C9⋊C9 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C9⋊C9 C32×C9 C3×C9 C33 C9 C32 C32 C3 # reps 1 1 2 2 6 6 3 1 6 2 18 6 3 6

Matrix representation of C9⋊(S3×C9) in GL8(𝔽19)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 11 0 0
,
 7 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0],[7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;

C9⋊(S3×C9) in GAP, Magma, Sage, TeX

C_9\rtimes (S_3\times C_9)
% in TeX

G:=Group("C9:(S3xC9)");
// GroupNames label

G:=SmallGroup(486,138);
// by ID

G=gap.SmallGroup(486,138);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,2169,453,3244,11669]);
// Polycyclic

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

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