metabelian, supersoluble, monomial
Aliases: C3⋊2D60, D30⋊2S3, C15⋊2D12, Dic3⋊D15, C6.5D30, C32⋊2D20, C30.25D6, C10.5S32, (C3×C15)⋊11D4, C6.5(S3×D5), (C6×D15)⋊6C2, C2.5(S3×D15), (C3×C6).5D10, C5⋊1(C3⋊D12), C15⋊7(C3⋊D4), C3⋊1(C3⋊D20), (C5×Dic3)⋊3S3, (C3×Dic3)⋊1D5, (Dic3×C15)⋊1C2, (C3×C30).19C22, (C2×C3⋊D15)⋊5C2, SmallGroup(360,81)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D60
G = < a,b,c | a3=b60=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 676 in 74 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, C20, D10, D12, C3⋊D4, C3×D5, D15, C30, C30, C3×Dic3, S3×C6, C2×C3⋊S3, D20, C3×C15, C5×Dic3, C60, C6×D5, D30, D30, C3⋊D12, C3×D15, C3⋊D15, C3×C30, C3⋊D20, D60, Dic3×C15, C6×D15, C2×C3⋊D15, C3⋊D60
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, D12, C3⋊D4, D15, S32, D20, S3×D5, D30, C3⋊D12, C3⋊D20, D60, S3×D15, C3⋊D60
►On 60 points
(1 21 41)(2 42 22)(3 23 43)(4 44 24)(5 25 45)(6 46 26)(7 27 47)(8 48 28)(9 29 49)(10 50 30)(11 31 51)(12 52 32)(13 33 53)(14 54 34)(15 35 55)(16 56 36)(17 37 57)(18 58 38)(19 39 59)(20 60 40)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)
G:=sub<Sym(60)| (1,21,41)(2,42,22)(3,23,43)(4,44,24)(5,25,45)(6,46,26)(7,27,47)(8,48,28)(9,29,49)(10,50,30)(11,31,51)(12,52,32)(13,33,53)(14,54,34)(15,35,55)(16,56,36)(17,37,57)(18,58,38)(19,39,59)(20,60,40), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)>;
G:=Group( (1,21,41)(2,42,22)(3,23,43)(4,44,24)(5,25,45)(6,46,26)(7,27,47)(8,48,28)(9,29,49)(10,50,30)(11,31,51)(12,52,32)(13,33,53)(14,54,34)(15,35,55)(16,56,36)(17,37,57)(18,58,38)(19,39,59)(20,60,40), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39) );
G=PermutationGroup([[(1,21,41),(2,42,22),(3,23,43),(4,44,24),(5,25,45),(6,46,26),(7,27,47),(8,48,28),(9,29,49),(10,50,30),(11,31,51),(12,52,32),(13,33,53),(14,54,34),(15,35,55),(16,56,36),(17,37,57),(18,58,38),(19,39,59),(20,60,40)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 12A | 12B | 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 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 30 | 90 | 2 | 2 | 4 | 6 | 2 | 2 | 2 | 2 | 4 | 30 | 30 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D5 | D6 | D10 | D12 | C3⋊D4 | D15 | D20 | D30 | D60 | S32 | S3×D5 | C3⋊D12 | C3⋊D20 | S3×D15 | C3⋊D60 |
| kernel | C3⋊D60 | Dic3×C15 | C6×D15 | C2×C3⋊D15 | C5×Dic3 | D30 | C3×C15 | C3×Dic3 | C30 | C3×C6 | C15 | C15 | Dic3 | C32 | C6 | C3 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 |
Matrix representation of C3⋊D60 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 59 | 59 | 0 | 0 | 0 | 0 |
| 2 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[59,2,0,0,0,0,59,32,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;
C3⋊D60 in GAP, Magma, Sage, TeX
C_3\rtimes D_{60} % in TeX
G:=Group("C3:D60"); // GroupNames label
G:=SmallGroup(360,81);
// by ID
G=gap.SmallGroup(360,81);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,201,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^3=b^60=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations