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## G = C3×Dic15order 180 = 22·32·5

### Direct product of C3 and Dic15

Aliases: C3×Dic15, C153C12, C30.1C6, C30.4S3, C6.4D15, C154Dic3, C322Dic5, C6.(C3×D5), (C3×C15)⋊8C4, C10.(C3×S3), C3⋊(C3×Dic5), C2.(C3×D15), (C3×C6).1D5, C52(C3×Dic3), (C3×C30).2C2, SmallGroup(180,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×Dic15
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C3×Dic15
 Lower central C15 — C3×Dic15
 Upper central C1 — C6

Generators and relations for C3×Dic15
G = < a,b,c | a3=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic15
On 60 points
Generators in S60
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 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 36 16 51)(2 35 17 50)(3 34 18 49)(4 33 19 48)(5 32 20 47)(6 31 21 46)(7 60 22 45)(8 59 23 44)(9 58 24 43)(10 57 25 42)(11 56 26 41)(12 55 27 40)(13 54 28 39)(14 53 29 38)(15 52 30 37)

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

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

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

C3×Dic15 is a maximal subgroup of
C3×D5×Dic3  C3×S3×Dic5  C6.D30  D62D15  C3⋊Dic30  D30.S3  Dic15⋊S3  D30⋊S3  C323Dic10  C12×D15

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 5A 5B 6A 6B 6C 6D 6E 10A 10B 12A 12B 12C 12D 15A ··· 15P 30A ··· 30P order 1 2 3 3 3 3 3 4 4 5 5 6 6 6 6 6 10 10 12 12 12 12 15 ··· 15 30 ··· 30 size 1 1 1 1 2 2 2 15 15 2 2 1 1 2 2 2 2 2 15 15 15 15 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C3 C4 C6 C12 S3 D5 Dic3 C3×S3 Dic5 C3×D5 D15 C3×Dic3 C3×Dic5 Dic15 C3×D15 C3×Dic15 kernel C3×Dic15 C3×C30 Dic15 C3×C15 C30 C15 C30 C3×C6 C15 C10 C32 C6 C6 C5 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 1 2 2 4 4 2 4 4 8 8

Matrix representation of C3×Dic15 in GL2(𝔽31) generated by

 5 0 0 5
,
 13 0 0 12
,
 0 30 1 0
G:=sub<GL(2,GF(31))| [5,0,0,5],[13,0,0,12],[0,1,30,0] >;

C3×Dic15 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{15}
% in TeX

G:=Group("C3xDic15");
// GroupNames label

G:=SmallGroup(180,15);
// by ID

G=gap.SmallGroup(180,15);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,483,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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