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## G = C3×C32⋊2D9order 486 = 2·35

### Direct product of C3 and C32⋊2D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C3×C32⋊2D9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C3×C32⋊C9 — C3×C32⋊2D9
 Lower central C32⋊C9 — C3×C32⋊2D9
 Upper central C1 — C32

Generators and relations for C3×C322D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1c, cd=dc, ce=ec, ede=d-1 >

Subgroups: 740 in 156 conjugacy classes, 30 normal (12 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, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C34, C322D9, C32×D9, C32×C3⋊S3, C3×C32⋊C9, C3×C322D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, He3⋊C2, C3×C3⋊S3, C322D9, C3×C9⋊S3, C3×He3⋊C2, C3×C322D9

Smallest permutation representation of C3×C322D9
On 54 points
Generators in S54
(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)
(2 41 30)(3 31 42)(5 44 33)(6 34 45)(8 38 36)(9 28 39)(10 54 24)(11 25 46)(13 48 27)(14 19 49)(16 51 21)(17 22 52)
(1 29 40)(2 30 41)(3 31 42)(4 32 43)(5 33 44)(6 34 45)(7 35 37)(8 36 38)(9 28 39)(10 24 54)(11 25 46)(12 26 47)(13 27 48)(14 19 49)(15 20 50)(16 21 51)(17 22 52)(18 23 53)
(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 11)(2 10)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 49)(38 48)(39 47)(40 46)(41 54)(42 53)(43 52)(44 51)(45 50)

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

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

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

63 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3Z 6A ··· 6H 9A ··· 9AA order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 27 1 ··· 1 2 ··· 2 6 ··· 6 27 ··· 27 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 C3×D9 He3⋊C2 C32⋊2D9 kernel C3×C32⋊2D9 C3×C32⋊C9 C32⋊2D9 C32⋊C9 C32×C9 C34 C3×C9 C33 C33 C32 C32 C3 # reps 1 1 2 2 3 1 6 9 2 18 12 6

Matrix representation of C3×C322D9 in GL5(𝔽19)

 7 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 7 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 13 14 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 9 0 0 0 0 5 17 0 0 0 0 0 0 1 0 0 0 9 3 10 0 0 5 2 16
,
 3 1 0 0 0 11 16 0 0 0 0 0 0 18 0 0 0 18 0 0 0 0 7 12 18

G:=sub<GL(5,GF(19))| [7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,12,0,0,0,0,11,0,0,0,0,0,1,0,13,0,0,0,11,14,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[9,5,0,0,0,0,17,0,0,0,0,0,0,9,5,0,0,1,3,2,0,0,0,10,16],[3,11,0,0,0,1,16,0,0,0,0,0,0,18,7,0,0,18,0,12,0,0,0,0,18] >;

C3×C322D9 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2D_9
% in TeX

G:=Group("C3xC3^2:2D9");
// GroupNames label

G:=SmallGroup(486,135);
// by ID

G=gap.SmallGroup(486,135);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,867,735,3244]);
// 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,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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