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G = C9×F5order 180 = 22·32·5

Direct product of C9 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9×F5, C5⋊C36, C452C4, C15.C12, D5.C18, C3.(C3×F5), (C3×F5).C3, (C3×D5).2C6, (C9×D5).2C2, SmallGroup(180,5)

Series: Derived Chief Lower central Upper central

C1C5 — C9×F5
C1C5C15C3×D5C9×D5 — C9×F5
C5 — C9×F5
C1C9

Generators and relations for C9×F5
 G = < a,b,c | a9=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

5C2
5C4
5C6
5C12
5C18
5C36

Smallest permutation representation of C9×F5
On 45 points
Generators in S45
(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)
(1 24 28 10 40)(2 25 29 11 41)(3 26 30 12 42)(4 27 31 13 43)(5 19 32 14 44)(6 20 33 15 45)(7 21 34 16 37)(8 22 35 17 38)(9 23 36 18 39)
(10 24 28 40)(11 25 29 41)(12 26 30 42)(13 27 31 43)(14 19 32 44)(15 20 33 45)(16 21 34 37)(17 22 35 38)(18 23 36 39)

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

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

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

45 conjugacy classes

class 1  2 3A3B4A4B 5 6A6B9A···9F12A12B12C12D15A15B18A···18F36A···36L45A···45F
order1233445669···912121212151518···1836···3645···45
size1511554551···15555445···55···54···4

45 irreducible representations

dim111111111444
type+++
imageC1C2C3C4C6C9C12C18C36F5C3×F5C9×F5
kernelC9×F5C9×D5C3×F5C45C3×D5F5C15D5C5C9C3C1
# reps1122264612126

Matrix representation of C9×F5 in GL5(𝔽181)

390000
0132000
0013200
0001320
0000132
,
10000
0180180180180
01000
00100
00010
,
190000
01000
00001
00100
0180180180180

G:=sub<GL(5,GF(181))| [39,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132,0,0,0,0,0,132],[1,0,0,0,0,0,180,1,0,0,0,180,0,1,0,0,180,0,0,1,0,180,0,0,0],[19,0,0,0,0,0,1,0,0,180,0,0,0,1,180,0,0,0,0,180,0,0,1,0,180] >;

C9×F5 in GAP, Magma, Sage, TeX

C_9\times F_5
% in TeX

G:=Group("C9xF5");
// GroupNames label

G:=SmallGroup(180,5);
// by ID

G=gap.SmallGroup(180,5);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,66,1804,614]);
// Polycyclic

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

Export

Subgroup lattice of C9×F5 in TeX

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