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G = C32×D5order 90 = 2·32·5

Direct product of C32 and D5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C32×D5, C152C6, C5⋊(C3×C6), (C3×C15)⋊3C2, SmallGroup(90,5)

Series: Derived Chief Lower central Upper central

C1C5 — C32×D5
C1C5C15C3×C15 — C32×D5
C5 — C32×D5
C1C32

Generators and relations for C32×D5
 G = < a,b,c,d | a3=b3=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
5C6
5C6
5C6
5C6
5C3×C6

Smallest permutation representation of C32×D5
On 45 points
Generators in S45
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 31 26)(7 32 27)(8 33 28)(9 34 29)(10 35 30)(11 36 16)(12 37 17)(13 38 18)(14 39 19)(15 40 20)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)

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

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

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

C32×D5 is a maximal subgroup of   C323F5

36 conjugacy classes

class 1  2 3A···3H5A5B6A···6H15A···15P
order123···3556···615···15
size151···1225···52···2

36 irreducible representations

dim111122
type+++
imageC1C2C3C6D5C3×D5
kernelC32×D5C3×C15C3×D5C15C32C3
# reps1188216

Matrix representation of C32×D5 in GL3(𝔽31) generated by

2500
010
001
,
100
050
005
,
100
0301
01713
,
3000
0300
0171
G:=sub<GL(3,GF(31))| [25,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,5],[1,0,0,0,30,17,0,1,13],[30,0,0,0,30,17,0,0,1] >;

C32×D5 in GAP, Magma, Sage, TeX

C_3^2\times D_5
% in TeX

G:=Group("C3^2xD5");
// GroupNames label

G:=SmallGroup(90,5);
// by ID

G=gap.SmallGroup(90,5);
# by ID

G:=PCGroup([4,-2,-3,-3,-5,1155]);
// Polycyclic

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

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Subgroup lattice of C32×D5 in TeX

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