metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10:4D6, D15:2D4, D6:4D10, Dic5:2D6, Dic3:2D10, D30:10C22, C30.27C23, C5:3(S3xD4), C3:3(D4xD5), C15:9(C2xD4), (C2xC10):5D6, (C2xC6):2D10, C5:D4:2S3, C3:D4:2D5, C3:D20:6C2, C5:D12:6C2, C22:4(S3xD5), D30.C2:5C2, (C2xC30):4C22, (C6xD5):4C22, (S3xC10):4C22, (C22xD15):6C2, C6.27(C22xD5), C10.27(C22xS3), (C5xDic3):2C22, (C3xDic5):2C22, (C2xS3xD5):6C2, C2.27(C2xS3xD5), (C3xC5:D4):4C2, (C5xC3:D4):4C2, SmallGroup(240,151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10:D6
G = < a,b,c,d | a10=b2=c6=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c-1 >
Subgroups: 600 in 108 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2xC4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2xC6, C2xC6, C15, C2xD4, Dic5, C20, D10, D10, C2xC10, C2xC10, C4xS3, D12, C3:D4, C3:D4, C3xD4, C22xS3, C5xS3, C3xD5, D15, D15, C30, C30, C4xD5, D20, C5:D4, C5:D4, C5xD4, C22xD5, S3xD4, C5xDic3, C3xDic5, S3xD5, C6xD5, S3xC10, D30, D30, C2xC30, D4xD5, D30.C2, C3:D20, C5:D12, C3xC5:D4, C5xC3:D4, C2xS3xD5, C22xD15, D10:D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, C22xS3, C22xD5, S3xD4, S3xD5, D4xD5, C2xS3xD5, D10:D6
►On 60 points
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)(41 43)(44 50)(45 49)(46 48)(51 59)(52 58)(53 57)(54 56)
(1 50 35 53 29 11)(2 41 36 54 30 12)(3 42 37 55 21 13)(4 43 38 56 22 14)(5 44 39 57 23 15)(6 45 40 58 24 16)(7 46 31 59 25 17)(8 47 32 60 26 18)(9 48 33 51 27 19)(10 49 34 52 28 20)
(1 11)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 50)(30 49)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 60)(39 59)(40 58)
G:=sub<Sym(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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,43)(44,50)(45,49)(46,48)(51,59)(52,58)(53,57)(54,56), (1,50,35,53,29,11)(2,41,36,54,30,12)(3,42,37,55,21,13)(4,43,38,56,22,14)(5,44,39,57,23,15)(6,45,40,58,24,16)(7,46,31,59,25,17)(8,47,32,60,26,18)(9,48,33,51,27,19)(10,49,34,52,28,20), (1,11)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,50)(30,49)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,60)(39,59)(40,58)>;
G:=Group( (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,43)(44,50)(45,49)(46,48)(51,59)(52,58)(53,57)(54,56), (1,50,35,53,29,11)(2,41,36,54,30,12)(3,42,37,55,21,13)(4,43,38,56,22,14)(5,44,39,57,23,15)(6,45,40,58,24,16)(7,46,31,59,25,17)(8,47,32,60,26,18)(9,48,33,51,27,19)(10,49,34,52,28,20), (1,11)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,50)(30,49)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,60)(39,59)(40,58) );
G=PermutationGroup([[(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,26),(22,25),(23,24),(27,30),(28,29),(31,38),(32,37),(33,36),(34,35),(39,40),(41,43),(44,50),(45,49),(46,48),(51,59),(52,58),(53,57),(54,56)], [(1,50,35,53,29,11),(2,41,36,54,30,12),(3,42,37,55,21,13),(4,43,38,56,22,14),(5,44,39,57,23,15),(6,45,40,58,24,16),(7,46,31,59,25,17),(8,47,32,60,26,18),(9,48,33,51,27,19),(10,49,34,52,28,20)], [(1,11),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,50),(30,49),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,60),(39,59),(40,58)]])
D10:D6 is a maximal subgroup of
D20:24D6 D20:29D6 S3xD4xD5 D30.C23 D20:14D6 D12:14D10 C15:2+ 1+4
D10:D6 is a maximal quotient of
Dic3:Dic10 D30.34D4 D30.35D4 D30:8Q8 Dic15:13D4 D30.Q8 Dic15:14D4 D30:D4 D10:4Dic6 D6:3Dic10 D30.6D4 D30:2D4 D30:12D4 Dic15.10D4 D30.27D4 D30:5D4 D15:D8 D30.8D4 Dic10:D6 D30.9D4 D20.10D6 Dic6:D10 D30.11D4 D12:5D10 D15:SD16 D60:C22 D15:Q16 C60.C23 D20.16D6 D20.17D6 D12.D10 D30.44D4 Dic15.19D4 D30:6D4 Dic15:3D4 D30:7D4 Dic15:4D4 Dic15:16D4 Dic15:17D4 D30.45D4 D30.16D4 Dic15:5D4 Dic15:18D4 D30:18D4 D30:19D4 Dic15.48D4 D30:8D4
33 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 | 12 | 15A | 15B | 20A | 20B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 6 | 10 | 15 | 15 | 30 | 2 | 6 | 10 | 2 | 2 | 2 | 4 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 20 | 4 | 4 | 12 | 12 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | S3xD4 | S3xD5 | D4xD5 | C2xS3xD5 | D10:D6 |
| kernel | D10:D6 | D30.C2 | C3:D20 | C5:D12 | C3xC5:D4 | C5xC3:D4 | C2xS3xD5 | C22xD15 | C5:D4 | D15 | C3:D4 | Dic5 | D10 | C2xC10 | Dic3 | D6 | C2xC6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of D10:D6 ►in GL6(F61)
| 17 | 18 | 0 | 0 | 0 | 0 |
| 44 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 44 | 1 | 0 | 0 | 0 | 0 |
| 17 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 20 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 20 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 44 | 43 | 0 | 0 | 0 | 0 |
| 16 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 20 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 20 |
| 0 | 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(6,GF(61))| [17,44,0,0,0,0,18,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,17,0,0,0,0,1,17,0,0,0,0,0,0,60,6,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,20,1,0,0,0,0,0,0,60,9,0,0,0,0,20,2],[44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,20,1,0,0,0,0,0,0,59,9,0,0,0,0,20,2] >;
D10:D6 in GAP, Magma, Sage, TeX
D_{10}\rtimes D_6 % in TeX
G:=Group("D10:D6"); // GroupNames label
G:=SmallGroup(240,151);
// by ID
G=gap.SmallGroup(240,151);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,116,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations