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## G = C5×3- 1+2order 135 = 33·5

### Direct product of C5 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C5×3- 1+2, C9⋊C15, C45⋊C3, C32.C15, C15.2C32, (C3×C15).C3, C3.2(C3×C15), SmallGroup(135,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×3- 1+2
 Chief series C1 — C3 — C15 — C45 — C5×3- 1+2
 Lower central C1 — C3 — C5×3- 1+2
 Upper central C1 — C15 — C5×3- 1+2

Generators and relations for C5×3- 1+2
G = < a,b,c | a5=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C5×3- 1+2
On 45 points
Generators in S45
(1 36 43 19 13)(2 28 44 20 14)(3 29 45 21 15)(4 30 37 22 16)(5 31 38 23 17)(6 32 39 24 18)(7 33 40 25 10)(8 34 41 26 11)(9 35 42 27 12)
(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)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(20 26 23)(21 24 27)(28 34 31)(29 32 35)(38 44 41)(39 42 45)

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

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

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

C5×3- 1+2 is a maximal subgroup of   D45⋊C3

55 conjugacy classes

 class 1 3A 3B 3C 3D 5A 5B 5C 5D 9A ··· 9F 15A ··· 15H 15I ··· 15P 45A ··· 45X order 1 3 3 3 3 5 5 5 5 9 ··· 9 15 ··· 15 15 ··· 15 45 ··· 45 size 1 1 1 3 3 1 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3

55 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + image C1 C3 C3 C5 C15 C15 3- 1+2 C5×3- 1+2 kernel C5×3- 1+2 C45 C3×C15 3- 1+2 C9 C32 C5 C1 # reps 1 6 2 4 24 8 2 8

Matrix representation of C5×3- 1+2 in GL3(𝔽181) generated by

 135 0 0 0 135 0 0 0 135
,
 0 1 0 0 0 132 1 0 0
,
 1 0 0 0 132 0 0 0 48
G:=sub<GL(3,GF(181))| [135,0,0,0,135,0,0,0,135],[0,0,1,1,0,0,0,132,0],[1,0,0,0,132,0,0,0,48] >;

C5×3- 1+2 in GAP, Magma, Sage, TeX

C_5\times 3_-^{1+2}
% in TeX

G:=Group("C5xES-(3,1)");
// GroupNames label

G:=SmallGroup(135,4);
// by ID

G=gap.SmallGroup(135,4);
# by ID

G:=PCGroup([4,-3,-3,-5,-3,180,385]);
// Polycyclic

G:=Group<a,b,c|a^5=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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