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G = C32×D5order 90 = 2·32·5

Direct product of C32 and D5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C32×D5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5
 Lower central C5 — C32×D5
 Upper central C1 — C32

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 >

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 ··· 3H 5A 5B 6A ··· 6H 15A ··· 15P order 1 2 3 ··· 3 5 5 6 ··· 6 15 ··· 15 size 1 5 1 ··· 1 2 2 5 ··· 5 2 ··· 2

36 irreducible representations

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

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

 25 0 0 0 1 0 0 0 1
,
 1 0 0 0 5 0 0 0 5
,
 1 0 0 0 30 1 0 17 13
,
 30 0 0 0 30 0 0 17 1
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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