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G = C32xF5order 180 = 22·32·5

Direct product of C32 and F5

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

Aliases: C32xF5, C15:2C12, C5:(C3xC12), (C3xC15):4C4, D5.(C3xC6), (C3xD5).3C6, (C32xD5).3C2, SmallGroup(180,20)

Series: Derived Chief Lower central Upper central

C1C5 — C32xF5
C1C5D5C3xD5C32xD5 — C32xF5
C5 — C32xF5
C1C32

Generators and relations for C32xF5
 G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 84 in 36 conjugacy classes, 24 normal (8 characteristic)
Quotients: C1, C2, C3, C4, C6, C32, C12, C3xC6, F5, C3xC12, C3xF5, C32xF5
5C2
5C4
5C6
5C6
5C6
5C6
5C12
5C12
5C12
5C12
5C3xC6
5C3xC12

Smallest permutation representation of C32xF5
On 45 points
Generators in S45
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)(32 33 35 34)(37 38 40 39)(42 43 45 44)

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

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

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

45 conjugacy classes

class 1  2 3A···3H4A4B 5 6A···6H12A···12P15A···15H
order123···34456···612···1215···15
size151···15545···55···54···4

45 irreducible representations

dim11111144
type+++
imageC1C2C3C4C6C12F5C3xF5
kernelC32xF5C32xD5C3xF5C3xC15C3xD5C15C32C3
# reps118281618

Matrix representation of C32xF5 in GL6(F61)

1300000
0470000
001000
000100
000010
000001
,
4700000
010000
001000
000100
000010
000001
,
100000
010000
0000060
0010060
0001060
0000160
,
5000000
0500000
000010
001000
000001
000100

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C32xF5 in GAP, Magma, Sage, TeX

C_3^2\times F_5
% in TeX

G:=Group("C3^2xF5");
// GroupNames label

G:=SmallGroup(180,20);
// by ID

G=gap.SmallGroup(180,20);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-5,90,1804,219]);
// Polycyclic

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

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Subgroup lattice of C32xF5 in TeX

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