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## G = C9×C32⋊C6order 486 = 2·35

### Direct product of C9 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C9×C32⋊C6
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9×He3 — C9×C32⋊C6
 Lower central C32 — C9×C32⋊C6
 Upper central C1 — C9

Generators and relations for C9×C32⋊C6
G = < a,b,c,d | a9=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 346 in 100 conjugacy classes, 33 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, He3, C33, C33, S3×C9, C32⋊C6, C3×C18, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C32×C9, C3×He3, C32⋊C18, S3×C3×C9, C3×C32⋊C6, C9×C3⋊S3, C9×He3, C9×C32⋊C6
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C32⋊C6, C3×C18, S3×C32, S3×C3×C9, C3×C32⋊C6, C9×C32⋊C6

Smallest permutation representation of C9×C32⋊C6
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 31 44)(2 32 45)(3 33 37)(4 34 38)(5 35 39)(6 36 40)(7 28 41)(8 29 42)(9 30 43)(10 53 23)(11 54 24)(12 46 25)(13 47 26)(14 48 27)(15 49 19)(16 50 20)(17 51 21)(18 52 22)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 15 38 49 31 22)(2 16 39 50 32 23)(3 17 40 51 33 24)(4 18 41 52 34 25)(5 10 42 53 35 26)(6 11 43 54 36 27)(7 12 44 46 28 19)(8 13 45 47 29 20)(9 14 37 48 30 21)

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3T 6A ··· 6H 9A ··· 9F 9G ··· 9L 9M ··· 9X 9Y ··· 9AP 18A ··· 18R order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 C32⋊C6 C3×C32⋊C6 C9×C32⋊C6 kernel C9×C32⋊C6 C9×He3 C32⋊C18 C3×C32⋊C6 C9×C3⋊S3 C32⋊C9 C32×C9 C3×He3 C32⋊C6 He3 C32×C9 C3×C9 C33 C32 C9 C3 C1 # reps 1 1 4 2 2 4 2 2 18 18 1 6 2 18 1 2 6

Matrix representation of C9×C32⋊C6 in GL6(𝔽19)

 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 7 0 15 0 0 0 0 0 18 0 0 0 0 11 12 0 0 0 0 0 0 11 13 0 0 0 0 0 8 7 0 0 0 0 18 0
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 0 0 1 0 13 0 0 0 8 0 8 0 0 0 11 1 18 1 0 13 0 0 0 8 0 8 0 0 0 11 1 18 0 0 0

G:=sub<GL(6,GF(19))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[7,0,0,0,0,0,0,0,11,0,0,0,15,18,12,0,0,0,0,0,0,11,0,0,0,0,0,13,8,18,0,0,0,0,7,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,0,1,8,11,0,0,0,0,0,1,0,0,0,13,8,18,1,8,11,0,0,0,0,0,1,0,0,0,13,8,18,0,0,0] >;

C9×C32⋊C6 in GAP, Magma, Sage, TeX

C_9\times C_3^2\rtimes C_6
% in TeX

G:=Group("C9xC3^2:C6");
// GroupNames label

G:=SmallGroup(486,98);
// by ID

G=gap.SmallGroup(486,98);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,3244,3250,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations

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