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## G = C15×Dic5order 300 = 22·3·52

### Direct product of C15 and Dic5

Aliases: C15×Dic5, C52C60, C10.C30, C154C20, C528C12, C30.8D5, C30.2C10, C2.(D5×C15), (C5×C15)⋊11C4, C6.2(C5×D5), (C5×C10).3C6, (C5×C30).3C2, C10.4(C3×D5), SmallGroup(300,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C15×Dic5
 Chief series C1 — C5 — C10 — C5×C10 — C5×C30 — C15×Dic5
 Lower central C5 — C15×Dic5
 Upper central C1 — C30

Generators and relations for C15×Dic5
G = < a,b,c | a15=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C15×Dic5
On 60 points
Generators in S60
(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)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 45 10 39 4 33 13 42 7 36)(2 31 11 40 5 34 14 43 8 37)(3 32 12 41 6 35 15 44 9 38)(16 46 22 52 28 58 19 49 25 55)(17 47 23 53 29 59 20 50 26 56)(18 48 24 54 30 60 21 51 27 57)
(1 50 33 23)(2 51 34 24)(3 52 35 25)(4 53 36 26)(5 54 37 27)(6 55 38 28)(7 56 39 29)(8 57 40 30)(9 58 41 16)(10 59 42 17)(11 60 43 18)(12 46 44 19)(13 47 45 20)(14 48 31 21)(15 49 32 22)

G:=sub<Sym(60)| (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,10,39,4,33,13,42,7,36)(2,31,11,40,5,34,14,43,8,37)(3,32,12,41,6,35,15,44,9,38)(16,46,22,52,28,58,19,49,25,55)(17,47,23,53,29,59,20,50,26,56)(18,48,24,54,30,60,21,51,27,57), (1,50,33,23)(2,51,34,24)(3,52,35,25)(4,53,36,26)(5,54,37,27)(6,55,38,28)(7,56,39,29)(8,57,40,30)(9,58,41,16)(10,59,42,17)(11,60,43,18)(12,46,44,19)(13,47,45,20)(14,48,31,21)(15,49,32,22)>;

G:=Group( (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)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,45,10,39,4,33,13,42,7,36)(2,31,11,40,5,34,14,43,8,37)(3,32,12,41,6,35,15,44,9,38)(16,46,22,52,28,58,19,49,25,55)(17,47,23,53,29,59,20,50,26,56)(18,48,24,54,30,60,21,51,27,57), (1,50,33,23)(2,51,34,24)(3,52,35,25)(4,53,36,26)(5,54,37,27)(6,55,38,28)(7,56,39,29)(8,57,40,30)(9,58,41,16)(10,59,42,17)(11,60,43,18)(12,46,44,19)(13,47,45,20)(14,48,31,21)(15,49,32,22) );

G=PermutationGroup([(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),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,45,10,39,4,33,13,42,7,36),(2,31,11,40,5,34,14,43,8,37),(3,32,12,41,6,35,15,44,9,38),(16,46,22,52,28,58,19,49,25,55),(17,47,23,53,29,59,20,50,26,56),(18,48,24,54,30,60,21,51,27,57)], [(1,50,33,23),(2,51,34,24),(3,52,35,25),(4,53,36,26),(5,54,37,27),(6,55,38,28),(7,56,39,29),(8,57,40,30),(9,58,41,16),(10,59,42,17),(11,60,43,18),(12,46,44,19),(13,47,45,20),(14,48,31,21),(15,49,32,22)])

120 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 5E ··· 5N 6A 6B 10A 10B 10C 10D 10E ··· 10N 12A 12B 12C 12D 15A ··· 15H 15I ··· 15AB 20A ··· 20H 30A ··· 30H 30I ··· 30AB 60A ··· 60P order 1 2 3 3 4 4 5 5 5 5 5 ··· 5 6 6 10 10 10 10 10 ··· 10 12 12 12 12 15 ··· 15 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 1 1 1 1 2 ··· 2 1 1 1 1 1 1 2 ··· 2 5 5 5 5 1 ··· 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 D5 Dic5 C3×D5 C5×D5 C3×Dic5 C5×Dic5 D5×C15 C15×Dic5 kernel C15×Dic5 C5×C30 C5×Dic5 C5×C15 C3×Dic5 C5×C10 C30 C52 Dic5 C15 C10 C5 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 2 2 4 8 4 8 16 16

Matrix representation of C15×Dic5 in GL2(𝔽31) generated by

 19 0 0 19
,
 23 0 0 27
,
 0 26 25 0
G:=sub<GL(2,GF(31))| [19,0,0,19],[23,0,0,27],[0,25,26,0] >;

C15×Dic5 in GAP, Magma, Sage, TeX

C_{15}\times {\rm Dic}_5
% in TeX

G:=Group("C15xDic5");
// GroupNames label

G:=SmallGroup(300,16);
// by ID

G=gap.SmallGroup(300,16);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-5,150,6004]);
// Polycyclic

G:=Group<a,b,c|a^15=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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