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

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

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

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

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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