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## G = C3×D5×Dic3order 360 = 23·32·5

### Direct product of C3, D5 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C3×D5×Dic3
 Chief series C1 — C5 — C15 — C30 — C3×C30 — D5×C3×C6 — C3×D5×Dic3
 Lower central C15 — C3×D5×Dic3
 Upper central C1 — C6

Generators and relations for C3×D5×Dic3
G = < a,b,c,d,e | a3=b5=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 244 in 74 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, Dic3, Dic3, C12, C2×C6, C15, C15, C3×C6, C3×C6, Dic5, C20, D10, C2×Dic3, C2×C12, C3×D5, C3×D5, C30, C30, C3×Dic3, C3×Dic3, C62, C4×D5, C3×C15, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, C6×Dic3, C32×D5, C3×C30, D5×Dic3, D5×C12, Dic3×C15, C3×Dic15, D5×C3×C6, C3×D5×Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, Dic3, C12, D6, C2×C6, C3×S3, D10, C2×Dic3, C2×C12, C3×D5, C3×Dic3, S3×C6, C4×D5, S3×D5, C6×D5, C6×Dic3, D5×Dic3, D5×C12, C3×S3×D5, C3×D5×Dic3

Smallest permutation representation of C3×D5×Dic3
On 60 points
Generators in S60
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)
(1 34 41 53 47)(2 35 42 54 48)(3 36 37 49 43)(4 31 38 50 44)(5 32 39 51 45)(6 33 40 52 46)(7 13 55 26 19)(8 14 56 27 20)(9 15 57 28 21)(10 16 58 29 22)(11 17 59 30 23)(12 18 60 25 24)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 29)(8 30)(9 25)(10 26)(11 27)(12 28)(13 58)(14 59)(15 60)(16 55)(17 56)(18 57)(19 22)(20 23)(21 24)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 40)(38 41)(39 42)
(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 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 10A 10B 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 6 ··· 6 10 10 12 12 12 12 12 12 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 5 5 1 1 2 2 2 3 3 15 15 2 2 1 1 2 2 2 5 5 5 5 10 ··· 10 2 2 3 3 3 3 15 15 15 15 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D5 Dic3 D6 C3×S3 D10 C3×D5 C3×Dic3 S3×C6 C4×D5 C6×D5 D5×C12 S3×D5 D5×Dic3 C3×S3×D5 C3×D5×Dic3 kernel C3×D5×Dic3 Dic3×C15 C3×Dic15 D5×C3×C6 D5×Dic3 C32×D5 C5×Dic3 Dic15 C6×D5 C3×D5 C6×D5 C3×Dic3 C3×D5 C30 D10 C3×C6 Dic3 D5 C10 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 2 2 1 2 2 4 4 2 4 4 8 2 2 4 4

Matrix representation of C3×D5×Dic3 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 60 1 0 0 16 44 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 45 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 13 34 0 0 0 47
,
 11 0 0 0 0 11 0 0 0 0 1 0 0 0 1 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[60,16,0,0,1,44,0,0,0,0,1,0,0,0,0,1],[1,45,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,13,0,0,0,34,47],[11,0,0,0,0,11,0,0,0,0,1,1,0,0,0,60] >;

C3×D5×Dic3 in GAP, Magma, Sage, TeX

C_3\times D_5\times {\rm Dic}_3
% in TeX

G:=Group("C3xD5xDic3");
// GroupNames label

G:=SmallGroup(360,58);
// by ID

G=gap.SmallGroup(360,58);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,79,730,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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