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G = C9⋊S3⋊C9  order 486 = 2·35

1st semidirect product of C9⋊S3 and C9 acting via C9/C3=C3

metabelian, supersoluble, monomial

Aliases: C9⋊S3⋊1C9, (C3×C9)⋊1D9, (C3×C9)⋊1C18, C3.1(C9×D9), C3.1(C9⋊C18), (C32×C9).1S3, (C32×C9).3C6, C32.8(C9⋊C6), C32.11(S3×C9), C33.73(C3×S3), C3.C92⋊1C2, C32.11(C3×D9), C3.1(C32⋊C18), C3.5(C32⋊D9), C32.24(C32⋊C6), (C3×C9⋊S3).1C3, SmallGroup(486,3)

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.C92 — 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=b3=c2=d9=1, dad-1=ab=ba, cac=a-1, cbc=b-1, bd=db, cd=dc >

Subgroups: 336 in 69 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C32×C9, C32×C9, C3×C9⋊S3, C9×C3⋊S3, C3.C92, C9⋊S3⋊C9
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3×S3, C3×D9, S3×C9, C32⋊C6, C9⋊C6, C9×D9, C32⋊C18, C32⋊D9, C9⋊C18, 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 38 31)(2 39 32)(3 40 33)(4 41 34)(5 42 35)(6 43 36)(7 44 28)(8 45 29)(9 37 30)(10 25 52)(11 26 53)(12 27 54)(13 19 46)(14 20 47)(15 21 48)(16 22 49)(17 23 50)(18 24 51)

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

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

63 conjugacy classes

class 1  2 3A3B3C···3N6A6B9A···9F9G···9AM18A···18F
order12333···3669···99···918···18
size127112···227273···36···627···27

63 irreducible representations

dim1111112222226666
type++++++
imageC1C2C3C6C9C18S3D9C3×S3C3×D9S3×C9C9×D9C32⋊C6C9⋊C6C32⋊C18C9⋊C18
kernelC9⋊S3⋊C9C3.C92C3×C9⋊S3C32×C9C9⋊S3C3×C9C32×C9C3×C9C33C32C32C3C32C32C3C3
# reps11226613266181224

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

50000000
04000000
005100000
000141000
000130000
000004013
00000003
000000115
,
10000000
01000000
00700000
00070000
00007000
000001100
000000110
000000011
,
018000000
180000000
00000100
00000010
00000001
00100000
00010000
00001000
,
10000000
01000000
009015000
00802000
0016110000
000009015
00000802
0000016110

G:=sub<GL(8,GF(19))| [5,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,10,14,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,3,15],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10] >;
 

C9⋊S3⋊C9 in GAP, Magma, Sage, TeX 

C_9\rtimes S_3\rtimes C_9
 
% in TeX
 
G:=Group("C9:S3:C9");
 
// GroupNames label
 
G:=SmallGroup(486,3);
 
// by ID
 
G=gap.SmallGroup(486,3);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,873,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^3=c^2=d^9=1,d*a*d^-1=a*b=b*a,c*a*c=a^-1,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

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