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## G = C9×C9⋊C6order 486 = 2·35

### Direct product of C9 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C9×C9⋊C6
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9×3- 1+2 — C9×C9⋊C6
 Lower central C9 — C9×C9⋊C6
 Upper central C1 — C9

Generators and relations for C9×C9⋊C6
G = < a,b,c | a9=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 238 in 82 conjugacy classes, 33 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, S3×C9, C9⋊C6, C3×C18, S3×C32, C92, C92, C32⋊C9, C9⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9×D9, C9⋊C18, S3×C3×C9, C3×C9⋊C6, C9×3- 1+2, C9×C9⋊C6
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C9⋊C6, C3×C18, S3×C32, S3×C3×C9, C3×C9⋊C6, C9×C9⋊C6

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

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 6A ··· 6H 9A ··· 9F 9G ··· 9L 9M ··· 9X 9Y ··· 9AY 18A ··· 18R order 1 2 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 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 C9⋊C6 C3×C9⋊C6 C9×C9⋊C6 kernel C9×C9⋊C6 C9×3- 1+2 C9×D9 C9⋊C18 C3×C9⋊C6 C92 C9⋊C9 C3×3- 1+2 C9⋊C6 3- 1+2 C32×C9 C3×C9 C33 C32 C9 C3 C1 # reps 1 1 2 4 2 2 4 2 18 18 1 6 2 18 1 2 6

Matrix representation of C9×C9⋊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
,
 18 11 8 3 14 16 18 0 8 14 17 0 6 0 1 14 17 7 0 0 0 0 0 1 0 0 0 7 0 0 0 0 0 0 7 0
,
 16 2 6 14 9 8 11 14 10 3 5 3 0 0 0 0 0 11 4 0 7 3 5 11 0 6 1 12 5 2 0 0 11 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],[18,18,6,0,0,0,11,0,0,0,0,0,8,8,1,0,0,0,3,14,14,0,7,0,14,17,17,0,0,7,16,0,7,1,0,0],[16,11,0,4,0,0,2,14,0,0,6,0,6,10,0,7,1,11,14,3,0,3,12,0,9,5,0,5,5,0,8,3,11,11,2,0] >;

C9×C9⋊C6 in GAP, Magma, Sage, TeX

C_9\times C_9\rtimes C_6
% in TeX

G:=Group("C9xC9:C6");
// GroupNames label

G:=SmallGroup(486,100);
// by ID

G=gap.SmallGroup(486,100);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,8104,3250,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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