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

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

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

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

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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