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G = C52⋊C9order 225 = 32·52

The semidirect product of C52 and C9 acting via C9/C3=C3

metabelian, soluble, monomial, A-group

Aliases: C52⋊C9, (C5×C15).C3, C3.(C52⋊C3), SmallGroup(225,3)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C9
C1C52C5×C15 — C52⋊C9
C52 — C52⋊C9
C1C3

Generators and relations for C52⋊C9
 G = < a,b,c | a5=b5=c9=1, cbc-1=ab=ba, cac-1=a3b2 >

3C5
3C5
25C9
3C15
3C15

Smallest permutation representation of C52⋊C9
On 45 points
Generators in S45
(2 17 25 41 33)(3 34 42 26 18)(5 11 19 44 36)(6 28 45 20 12)(8 14 22 38 30)(9 31 39 23 15)
(1 40 16 32 24)(2 41 17 33 25)(3 34 42 26 18)(4 43 10 35 27)(5 44 11 36 19)(6 28 45 20 12)(7 37 13 29 21)(8 38 14 30 22)(9 31 39 23 15)
(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)

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

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

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

C52⋊C9 is a maximal subgroup of   C52⋊D9  C52⋊C18

33 conjugacy classes

class 1 3A3B5A···5H9A···9F15A···15P
order1335···59···915···15
size1113···325···253···3

33 irreducible representations

dim11133
type+
imageC1C3C9C52⋊C3C52⋊C9
kernelC52⋊C9C5×C15C52C3C1
# reps126816

Matrix representation of C52⋊C9 in GL4(𝔽181) generated by

1000
0100
0511350
068059
,
1000
012500
01741350
000125
,
39000
0581240
0581231
0591230
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,51,68,0,0,135,0,0,0,0,59],[1,0,0,0,0,125,174,0,0,0,135,0,0,0,0,125],[39,0,0,0,0,58,58,59,0,124,123,123,0,0,1,0] >;

C52⋊C9 in GAP, Magma, Sage, TeX

C_5^2\rtimes C_9
% in TeX

G:=Group("C5^2:C9");
// GroupNames label

G:=SmallGroup(225,3);
// by ID

G=gap.SmallGroup(225,3);
# by ID

G:=PCGroup([4,-3,-3,-5,5,12,1730,2739]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^9=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^3*b^2>;
// generators/relations

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Subgroup lattice of C52⋊C9 in TeX

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