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## G = C32×F5order 180 = 22·32·5

### Direct product of C32 and F5

Aliases: C32×F5, C152C12, C5⋊(C3×C12), (C3×C15)⋊4C4, D5.(C3×C6), (C3×D5).3C6, (C32×D5).3C2, SmallGroup(180,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C32×F5
 Chief series C1 — C5 — D5 — C3×D5 — C32×D5 — C32×F5
 Lower central C5 — C32×F5
 Upper central C1 — C32

Generators and relations for C32×F5
G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of C32×F5
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 ··· 3H 4A 4B 5 6A ··· 6H 12A ··· 12P 15A ··· 15H order 1 2 3 ··· 3 4 4 5 6 ··· 6 12 ··· 12 15 ··· 15 size 1 5 1 ··· 1 5 5 4 5 ··· 5 5 ··· 5 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C3 C4 C6 C12 F5 C3×F5 kernel C32×F5 C32×D5 C3×F5 C3×C15 C3×D5 C15 C32 C3 # reps 1 1 8 2 8 16 1 8

Matrix representation of C32×F5 in GL6(𝔽61)

 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 0 0 60 0 0 1 0 0 60 0 0 0 1 0 60 0 0 0 0 1 60
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

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] >;

C32×F5 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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