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## G = C9×C9⋊S3order 486 = 2·35

### Direct product of C9 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C9×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C92 — C9×C9⋊S3
 Lower central C3×C9 — C9×C9⋊S3
 Upper central C1 — C9

Generators and relations for C9×C9⋊S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 354 in 108 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C92, C92, C32×C9, C32×C9, C9×D9, C3×C9⋊S3, C9×C3⋊S3, C3×C92, C9×C9⋊S3
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3×S3, C3⋊S3, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C9×D9, C3×C9⋊S3, C9×C3⋊S3, C9×C9⋊S3

Smallest permutation representation of C9×C9⋊S3
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 31 7 38 28 4 44 34)(2 42 32 8 39 29 5 45 35)(3 43 33 9 40 30 6 37 36)(10 48 22 13 51 25 16 54 19)(11 49 23 14 52 26 17 46 20)(12 50 24 15 53 27 18 47 21)
(1 36 39)(2 28 40)(3 29 41)(4 30 42)(5 31 43)(6 32 44)(7 33 45)(8 34 37)(9 35 38)(10 26 50)(11 27 51)(12 19 52)(13 20 53)(14 21 54)(15 22 46)(16 23 47)(17 24 48)(18 25 49)
(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 46)(44 47)(45 48)

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

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

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

135 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9F 9G ··· 9DG 18A ··· 18F order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 ··· 2 27 27 1 ··· 1 2 ··· 2 27 ··· 27

135 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 C9 C18 S3 S3 D9 C3×S3 C3×S3 S3×C9 C3×D9 S3×C9 C9×D9 kernel C9×C9⋊S3 C3×C92 C3×C9⋊S3 C32×C9 C9⋊S3 C3×C9 C92 C32×C9 C3×C9 C3×C9 C33 C9 C32 C32 C3 # reps 1 1 2 2 6 6 3 1 9 6 2 18 18 6 54

Matrix representation of C9×C9⋊S3 in GL4(𝔽19) generated by

 9 0 0 0 0 9 0 0 0 0 6 0 0 0 0 6
,
 5 0 0 0 0 4 0 0 0 0 16 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 0 7 0 0 0 0 11
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(19))| [9,0,0,0,0,9,0,0,0,0,6,0,0,0,0,6],[5,0,0,0,0,4,0,0,0,0,16,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C9×C9⋊S3 in GAP, Magma, Sage, TeX

C_9\times C_9\rtimes S_3
% in TeX

G:=Group("C9xC9:S3");
// GroupNames label

G:=SmallGroup(486,133);
// by ID

G=gap.SmallGroup(486,133);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,453,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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