direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×D5×Dic3, C30.28D6, Dic15⋊2C6, C3⋊3(D5×C12), C6.1(C6×D5), C5⋊2(C6×Dic3), C32⋊9(C4×D5), C15⋊4(C2×C12), (C3×D5)⋊1C12, C10.1(S3×C6), C30.1(C2×C6), (C6×D5).9S3, (C6×D5).1C6, C6.28(S3×D5), (C5×Dic3)⋊1C6, D10.2(C3×S3), C15⋊8(C2×Dic3), (C3×C6).13D10, (C32×D5)⋊3C4, (C3×Dic15)⋊2C2, (Dic3×C15)⋊6C2, (C3×C30).1C22, C2.1(C3×S3×D5), (D5×C3×C6).1C2, (C3×C15)⋊13(C2×C4), SmallGroup(360,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3×D5×Dic3 |
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
►On 60 points
(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