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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

C1C3 — C5×3- 1+2
C1C3C15C45 — C5×3- 1+2
C1C3 — C5×3- 1+2
C1C15 — 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 >

3C3
3C15

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 3A3B3C3D5A5B5C5D9A···9F15A···15H15I···15P45A···45X
order1333355559···915···1515···1545···45
size1113311113···31···13···33···3

55 irreducible representations

dim11111133
type+
imageC1C3C3C5C15C153- 1+2C5×3- 1+2
kernelC5×3- 1+2C45C3×C153- 1+2C9C32C5C1
# reps162424828

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

13500
01350
00135
,
010
00132
100
,
100
01320
0048
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

Export

Subgroup lattice of C5×3- 1+2 in TeX

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