direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D12, C20⋊1D6, D6⋊1D10, C12⋊4D10, D60⋊10C2, C60⋊3C22, Dic5⋊3D6, D10.18D6, D30⋊1C22, C30.12C23, C3⋊1(D4×D5), C4⋊2(S3×D5), C5⋊1(C2×D12), C15⋊1(C2×D4), (C4×D5)⋊3S3, (C3×D5)⋊1D4, (D5×C12)⋊3C2, (C5×D12)⋊3C2, C5⋊D12⋊3C2, (S3×C10)⋊1C22, C6.12(C22×D5), C10.12(C22×S3), (C3×Dic5)⋊4C22, (C6×D5).14C22, (C2×S3×D5)⋊1C2, C2.15(C2×S3×D5), SmallGroup(240,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D12
G = < a,b,c,d | a5=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 608 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, C12, C12, D6, D6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10, C2×C10, D12, D12, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C2×D12, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, D4×D5, C5⋊D12, D5×C12, C5×D12, D60, C2×S3×D5, D5×D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C22×D5, C2×D12, S3×D5, D4×D5, C2×S3×D5, D5×D12
►On 60 points
(1 43 58 33 17)(2 44 59 34 18)(3 45 60 35 19)(4 46 49 36 20)(5 47 50 25 21)(6 48 51 26 22)(7 37 52 27 23)(8 38 53 28 24)(9 39 54 29 13)(10 40 55 30 14)(11 41 56 31 15)(12 42 57 32 16)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 55)(50 56)(51 57)(52 58)(53 59)(54 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)
G:=sub<Sym(60)| (1,43,58,33,17)(2,44,59,34,18)(3,45,60,35,19)(4,46,49,36,20)(5,47,50,25,21)(6,48,51,26,22)(7,37,52,27,23)(8,38,53,28,24)(9,39,54,29,13)(10,40,55,30,14)(11,41,56,31,15)(12,42,57,32,16), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58)>;
G:=Group( (1,43,58,33,17)(2,44,59,34,18)(3,45,60,35,19)(4,46,49,36,20)(5,47,50,25,21)(6,48,51,26,22)(7,37,52,27,23)(8,38,53,28,24)(9,39,54,29,13)(10,40,55,30,14)(11,41,56,31,15)(12,42,57,32,16), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,54)(50,53)(51,52)(55,60)(56,59)(57,58) );
G=PermutationGroup([[(1,43,58,33,17),(2,44,59,34,18),(3,45,60,35,19),(4,46,49,36,20),(5,47,50,25,21),(6,48,51,26,22),(7,37,52,27,23),(8,38,53,28,24),(9,39,54,29,13),(10,40,55,30,14),(11,41,56,31,15),(12,42,57,32,16)], [(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(49,54),(50,53),(51,52),(55,60),(56,59),(57,58)]])
D5×D12 is a maximal subgroup of
D60⋊C4 D12⋊F5 C24⋊D10 D24⋊D5 Dic10⋊3D6 D20⋊D6 D60⋊3C4 D20⋊26D6 D20⋊29D6 S3×D4×D5 D12⋊14D10 D20⋊17D6
D5×D12 is a maximal quotient of
C24⋊D10 D24⋊D5 Dic60⋊C2 C24.2D10 C40.31D6 D24⋊7D5 D120⋊C2 Dic5.8D12 Dic5⋊4D12 D10.16D12 D10.17D12 Dic5⋊D12 D6⋊2Dic10 D60⋊17C4 D30⋊2Q8 D30⋊D4 D10⋊D12 C60⋊D4 C12⋊7D20 C20⋊D12 C60⋊Q8 D30⋊4D4 D30⋊5D4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 6 | 6 | 30 | 30 | 2 | 2 | 10 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D12 | S3×D5 | D4×D5 | C2×S3×D5 | D5×D12 |
| kernel | D5×D12 | C5⋊D12 | D5×C12 | C5×D12 | D60 | C2×S3×D5 | C4×D5 | C3×D5 | D12 | Dic5 | C20 | D10 | C12 | D6 | D5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of D5×D12 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 43 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 43 |
| 0 | 0 | 0 | 60 |
| 23 | 38 | 0 | 0 |
| 23 | 46 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 23 | 38 | 0 | 0 |
| 15 | 38 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,43,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,43,60],[23,23,0,0,38,46,0,0,0,0,60,0,0,0,0,60],[23,15,0,0,38,38,0,0,0,0,60,0,0,0,0,60] >;
D5×D12 in GAP, Magma, Sage, TeX
D_5\times D_{12} % in TeX
G:=Group("D5xD12"); // GroupNames label
G:=SmallGroup(240,136);
// by ID
G=gap.SmallGroup(240,136);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations