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G = C3×C32⋊D9  order 486 = 2·35

Direct product of C3 and C32⋊D9

direct product, metabelian, supersoluble, monomial

Aliases: C3×C32⋊D9, C33⋊3D9, C34.3S3, C9⋊S3⋊5C32, (C32×C9)⋊6C6, C32⋊2(C3×D9), C32⋊C9⋊19C6, C3.1(C32×D9), C32.9(C9⋊C6), C33.30(C3×S3), C32.29(S3×C32), C32.41(C32⋊C6), (C3×C9⋊S3)⋊1C3, (C3×C9)⋊9(C3×C6), C3.1(C3×C9⋊C6), (C3×C32⋊C9)⋊4C2, C3.1(C3×C32⋊C6), SmallGroup(486,94)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C3×C32⋊D9
Chief series
C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C32⋊C9 — C3×C32⋊D9
Lower central
C3×C9 — C3×C32⋊D9
Upper central
C1 — C3

Generators and relations for C3×C32⋊D9
 G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 708 in 138 conjugacy classes, 32 normal (16 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, C3×D9, C9⋊S3, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C34, C32⋊D9, C3×C9⋊S3, C32×C3⋊S3, C3×C32⋊C9, C3×C32⋊D9
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C3×D9, C32⋊C6, C9⋊C6, S3×C32, C32⋊D9, C32×D9, C3×C32⋊C6, C3×C9⋊C6, C3×C32⋊D9

Smallest permutation representation of C3×C32⋊D9
►On 54 points
Generators in S54
(1 33 41)(2 34 42)(3 35 43)(4 36 44)(5 28 45)(6 29 37)(7 30 38)(8 31 39)(9 32 40)(10 21 54)(11 22 46)(12 23 47)(13 24 48)(14 25 49)(15 26 50)(16 27 51)(17 19 52)(18 20 53)

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

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

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

63 conjugacy classes

class 1  2 3A3B3C···3N3O···3T3U···3Z6A···6H9A···9AA
order12333···33···33···36···69···9
size127112···23···36···627···276···6

63 irreducible representations

dim11111122226666
type++++++
imageC1C2C3C3C6C6S3D9C3×S3C3×D9C32⋊C6C9⋊C6C3×C32⋊C6C3×C9⋊C6
kernelC3×C32⋊D9C3×C32⋊C9C32⋊D9C3×C9⋊S3C32⋊C9C32×C9C34C33C33C32C32C32C3C3
# reps116262138241224

Matrix representation of C3×C32⋊D9 ►in GL8(𝔽19)

70000000
07000000
001100000
000110000
000011000
000001100
000000110
000000011
,
10000000
01000000
00100000
005110000
001507000
00000100
000005110
000001507
,
10000000
01000000
001100000
000110000
000011000
00000700
00000070
00000007
,
90000000
017000000
009015000
00002000
000710000
0000017150
000000211
000000100
,
02000000
100000000
0000017150
000000211
000000100
009015000
00002000
000710000

G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,5,15,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,5,15,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7],[9,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,15,2,10,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,15,2,10,0,0,0,0,0,0,11,0],[0,10,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,15,2,10,0,0,17,0,0,0,0,0,0,0,15,2,10,0,0,0,0,0,0,11,0,0,0,0] >;
 

C3×C32⋊D9 in GAP, Magma, Sage, TeX 

C_3\times C_3^2\rtimes D_9
 
% in TeX
 
G:=Group("C3xC3^2:D9");
 
// GroupNames label
 
G:=SmallGroup(486,94);
 
// by ID
 
G=gap.SmallGroup(486,94);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,873,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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