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## G = C3×C32⋊D9order 486 = 2·35

### Direct product of C3 and C32⋊D9

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 3A 3B 3C ··· 3N 3O ··· 3T 3U ··· 3Z 6A ··· 6H 9A ··· 9AA order 1 2 3 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 1 1 2 ··· 2 3 ··· 3 6 ··· 6 27 ··· 27 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 6 6 6 6 type + + + + + + image C1 C2 C3 C3 C6 C6 S3 D9 C3×S3 C3×D9 C32⋊C6 C9⋊C6 C3×C32⋊C6 C3×C9⋊C6 kernel C3×C32⋊D9 C3×C32⋊C9 C32⋊D9 C3×C9⋊S3 C32⋊C9 C32×C9 C34 C33 C33 C32 C32 C32 C3 C3 # reps 1 1 6 2 6 2 1 3 8 24 1 2 2 4

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

 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 0 0 0 0 0 0 0 5 11 0 0 0 0 0 0 15 0 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 11 0 0 0 0 0 0 15 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 15 0 0 0 0 0 0 0 2 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 17 15 0 0 0 0 0 0 0 2 11 0 0 0 0 0 0 10 0
,
 0 2 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 17 15 0 0 0 0 0 0 0 2 11 0 0 0 0 0 0 10 0 0 0 9 0 15 0 0 0 0 0 0 0 2 0 0 0 0 0 0 7 10 0 0 0

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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