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## G = D9×3- 1+2order 486 = 2·35

### Direct product of D9 and 3- 1+2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — D9×3- 1+2
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9×3- 1+2 — D9×3- 1+2
 Lower central C9 — C3×C9 — D9×3- 1+2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for D9×3- 1+2
G = < a,b,c,d | a9=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 238 in 73 conjugacy classes, 28 normal (16 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×D9, C3×D9, S3×C9, C2×3- 1+2, S3×C32, C92, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9×D9, C32×D9, S3×3- 1+2, C9×3- 1+2, D9×3- 1+2
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, 3- 1+2, C3×D9, C2×3- 1+2, S3×C32, C32×D9, S3×3- 1+2, D9×3- 1+2

Smallest permutation representation of D9×3- 1+2
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 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 45)(18 44)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)
(1 23 14 4 26 17 7 20 11)(2 24 15 5 27 18 8 21 12)(3 25 16 6 19 10 9 22 13)(28 46 37 34 52 43 31 49 40)(29 47 38 35 53 44 32 50 41)(30 48 39 36 54 45 33 51 42)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)

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

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

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

66 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 9A ··· 9I 9J ··· 9O 9P ··· 9AS 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 2 2 2 3 3 6 6 9 9 27 27 2 ··· 2 3 ··· 3 6 ··· 6 27 ··· 27

66 irreducible representations

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

Matrix representation of D9×3- 1+2 in GL5(𝔽19)

 0 1 0 0 0 18 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 9 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 7 0 0 7 0 0 0 0 0 11 0
,
 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 7

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,9,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,11,0,0,7,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7] >;

D9×3- 1+2 in GAP, Magma, Sage, TeX

D_9\times 3_-^{1+2}
% in TeX

G:=Group("D9xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,101);
// by ID

G=gap.SmallGroup(486,101);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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