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## G = Dic3×C15order 180 = 22·32·5

### Direct product of C15 and Dic3

Aliases: Dic3×C15, C3⋊C60, C6.C30, C155C12, C30.5C6, C30.8S3, C322C20, C2.(S3×C15), (C3×C15)⋊10C4, C6.4(C5×S3), C10.2(C3×S3), (C3×C6).1C10, (C3×C30).4C2, SmallGroup(180,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C15
 Chief series C1 — C3 — C6 — C30 — C3×C30 — Dic3×C15
 Lower central C3 — Dic3×C15
 Upper central C1 — C30

Generators and relations for Dic3×C15
G = < a,b,c | a15=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C15
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 16 11 26 6 21)(2 17 12 27 7 22)(3 18 13 28 8 23)(4 19 14 29 9 24)(5 20 15 30 10 25)(31 58 36 48 41 53)(32 59 37 49 42 54)(33 60 38 50 43 55)(34 46 39 51 44 56)(35 47 40 52 45 57)
(1 40 26 57)(2 41 27 58)(3 42 28 59)(4 43 29 60)(5 44 30 46)(6 45 16 47)(7 31 17 48)(8 32 18 49)(9 33 19 50)(10 34 20 51)(11 35 21 52)(12 36 22 53)(13 37 23 54)(14 38 24 55)(15 39 25 56)

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

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

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

Dic3×C15 is a maximal subgroup of   C6.D30  C3⋊D60  C3⋊Dic30  S3×C60

90 conjugacy classes

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

90 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 S3 Dic3 C3×S3 C5×S3 C3×Dic3 C5×Dic3 S3×C15 Dic3×C15 kernel Dic3×C15 C3×C30 C5×Dic3 C3×C15 C3×Dic3 C30 C3×C6 C15 Dic3 C32 C6 C3 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 1 2 4 2 4 8 8

Matrix representation of Dic3×C15 in GL2(𝔽31) generated by

 20 0 0 20
,
 26 0 0 6
,
 0 30 1 0
G:=sub<GL(2,GF(31))| [20,0,0,20],[26,0,0,6],[0,1,30,0] >;

Dic3×C15 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{15}
% in TeX

G:=Group("Dic3xC15");
// GroupNames label

G:=SmallGroup(180,14);
// by ID

G=gap.SmallGroup(180,14);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-3,150,3004]);
// Polycyclic

G:=Group<a,b,c|a^15=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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