metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2Dic3, D10.14D6, (C2×C6)⋊2F5, (C2×C30)⋊1C4, (C6×D5)⋊4C4, C5⋊(C6.D4), (C3×D5).5D4, C22⋊2(C3⋊F5), C3⋊2(C22⋊F5), C6.14(C2×F5), C15⋊2(C22⋊C4), C30.14(C2×C4), (C2×C10)⋊4Dic3, D5.3(C3⋊D4), C10.7(C2×Dic3), (C22×D5).3S3, (C6×D5).21C22, (C2×C3⋊F5)⋊3C2, C2.7(C2×C3⋊F5), (D5×C2×C6).4C2, SmallGroup(240,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.D6
G = < a,b,c,d | a10=b2=c6=1, d2=a4b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a5c-1 >
Subgroups: 320 in 68 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, C2×C6, C2×C6, C15, C22⋊C4, F5, D10, D10, C2×C10, C2×Dic3, C22×C6, C3×D5, C3×D5, C30, C30, C2×F5, C22×D5, C6.D4, C3⋊F5, C6×D5, C6×D5, C2×C30, C22⋊F5, C2×C3⋊F5, D5×C2×C6, D10.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C2×C3⋊F5, D10.D6
Character table of D10.D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 15A | 15B | 30A | 30B | 30C | 30D | 30E | 30F | |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 30 | 30 | 30 | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -2 | -2 | 2 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -2 | -2 | 2 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 0 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | -1 | 0 | 0 | -2 | -1 | -1 | -√-3 | -√-3 | √-3 | 1 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | -2 | 0 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | 0 | 0 | -2 | -1 | -1 | -√-3 | -√-3 | √-3 | 1 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | 0 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | 0 | 0 | -2 | -1 | -1 | √-3 | √-3 | -√-3 | 1 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | 0 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 2 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | -1 | 0 | 0 | -2 | -1 | -1 | √-3 | √-3 | -√-3 | 1 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ20 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | -1 | -1 | √5 | -√5 | -√5 | 1 | 1 | √5 | orthogonal lifted from C22⋊F5 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | -1 | -1 | -√5 | √5 | √5 | 1 | 1 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ23 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 1-√-15/2 | 1+√-15/2 | -ζ32+ζ53+ζ52 | -ζ32+ζ54+ζ5 | -ζ3+ζ54+ζ5 | -1+√-15/2 | -1-√-15/2 | -ζ3+ζ53+ζ52 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 1+√-15/2 | 1-√-15/2 | -ζ32+ζ54+ζ5 | -ζ32+ζ53+ζ52 | -ζ3+ζ53+ζ52 | -1-√-15/2 | -1+√-15/2 | -ζ3+ζ54+ζ5 | complex faithful |
| ρ26 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 1+√-15/2 | 1-√-15/2 | -ζ3+ζ53+ζ52 | -ζ3+ζ54+ζ5 | -ζ32+ζ54+ζ5 | -1-√-15/2 | -1+√-15/2 | -ζ32+ζ53+ζ52 | complex faithful |
| ρ28 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1+√-15/2 | 1-√-15/2 | -1-√-15/2 | -1+√-15/2 | -1-√-15/2 | 1+√-15/2 | 1-√-15/2 | -1+√-15/2 | complex lifted from C2×C3⋊F5 |
| ρ29 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1-√-15/2 | 1+√-15/2 | -1+√-15/2 | -1-√-15/2 | -1+√-15/2 | 1-√-15/2 | 1+√-15/2 | -1-√-15/2 | complex lifted from C2×C3⋊F5 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 1-√-15/2 | 1+√-15/2 | -ζ3+ζ54+ζ5 | -ζ3+ζ53+ζ52 | -ζ32+ζ53+ζ52 | -1+√-15/2 | -1-√-15/2 | -ζ32+ζ54+ζ5 | complex faithful |
►On 60 points
Generators in S60
(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 20)(12 19)(13 18)(14 17)(15 16)(21 28)(22 27)(23 26)(24 25)(29 30)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 54)(52 53)(55 60)(56 59)(57 58)
(1 50 35 53 25 11)(2 41 36 54 26 12)(3 42 37 55 27 13)(4 43 38 56 28 14)(5 44 39 57 29 15)(6 45 40 58 30 16)(7 46 31 59 21 17)(8 47 32 60 22 18)(9 48 33 51 23 19)(10 49 34 52 24 20)
(1 11 6 16)(2 14 5 13)(3 17 4 20)(7 19 10 18)(8 12 9 15)(21 48 24 47)(22 41 23 44)(25 50 30 45)(26 43 29 42)(27 46 28 49)(31 51 34 60)(32 54 33 57)(35 53 40 58)(36 56 39 55)(37 59 38 52)
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)(11,20)(12,19)(13,18)(14,17)(15,16)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58), (1,50,35,53,25,11)(2,41,36,54,26,12)(3,42,37,55,27,13)(4,43,38,56,28,14)(5,44,39,57,29,15)(6,45,40,58,30,16)(7,46,31,59,21,17)(8,47,32,60,22,18)(9,48,33,51,23,19)(10,49,34,52,24,20), (1,11,6,16)(2,14,5,13)(3,17,4,20)(7,19,10,18)(8,12,9,15)(21,48,24,47)(22,41,23,44)(25,50,30,45)(26,43,29,42)(27,46,28,49)(31,51,34,60)(32,54,33,57)(35,53,40,58)(36,56,39,55)(37,59,38,52)>;
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)(11,20)(12,19)(13,18)(14,17)(15,16)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58), (1,50,35,53,25,11)(2,41,36,54,26,12)(3,42,37,55,27,13)(4,43,38,56,28,14)(5,44,39,57,29,15)(6,45,40,58,30,16)(7,46,31,59,21,17)(8,47,32,60,22,18)(9,48,33,51,23,19)(10,49,34,52,24,20), (1,11,6,16)(2,14,5,13)(3,17,4,20)(7,19,10,18)(8,12,9,15)(21,48,24,47)(22,41,23,44)(25,50,30,45)(26,43,29,42)(27,46,28,49)(31,51,34,60)(32,54,33,57)(35,53,40,58)(36,56,39,55)(37,59,38,52) );
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),(11,20),(12,19),(13,18),(14,17),(15,16),(21,28),(22,27),(23,26),(24,25),(29,30),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,54),(52,53),(55,60),(56,59),(57,58)], [(1,50,35,53,25,11),(2,41,36,54,26,12),(3,42,37,55,27,13),(4,43,38,56,28,14),(5,44,39,57,29,15),(6,45,40,58,30,16),(7,46,31,59,21,17),(8,47,32,60,22,18),(9,48,33,51,23,19),(10,49,34,52,24,20)], [(1,11,6,16),(2,14,5,13),(3,17,4,20),(7,19,10,18),(8,12,9,15),(21,48,24,47),(22,41,23,44),(25,50,30,45),(26,43,29,42),(27,46,28,49),(31,51,34,60),(32,54,33,57),(35,53,40,58),(36,56,39,55),(37,59,38,52)]])
D10.D6 is a maximal subgroup of
D10.D12 D10.4D12 (C2×C60)⋊C4 C3⋊(C23⋊F5) C22⋊F5.S3 F5×C3⋊D4 S3×C22⋊F5 (C2×C12)⋊6F5 D4×C3⋊F5
D10.D6 is a maximal quotient of
(C2×C60)⋊C4 C30.7M4(2) (C2×C60).C4 D10.10D12 D20⋊Dic3 Dic10⋊Dic3 Dic10⋊2Dic3 D20⋊2Dic3 C3⋊(C23⋊F5) C30.22M4(2) C5⋊(C12.D4)
Matrix representation of D10.D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 0 | 60 | 0 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 0 | 1 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 29 | 55 | 0 | 0 | 0 | 0 |
| 18 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 6 | 0 | 55 |
| 0 | 0 | 0 | 34 | 6 | 55 |
| 0 | 0 | 55 | 6 | 34 | 0 |
| 0 | 0 | 55 | 0 | 6 | 28 |
| 14 | 56 | 0 | 0 | 0 | 0 |
| 39 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 55 | 6 |
| 0 | 0 | 6 | 0 | 55 | 34 |
| 0 | 0 | 34 | 55 | 0 | 6 |
| 0 | 0 | 6 | 55 | 28 | 0 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,60,0,0,0,0,0,1],[29,18,0,0,0,0,55,32,0,0,0,0,0,0,28,0,55,55,0,0,6,34,6,0,0,0,0,6,34,6,0,0,55,55,0,28],[14,39,0,0,0,0,56,47,0,0,0,0,0,0,0,6,34,6,0,0,28,0,55,55,0,0,55,55,0,28,0,0,6,34,6,0] >;
D10.D6 in GAP, Magma, Sage, TeX
D_{10}.D_6 % in TeX
G:=Group("D10.D6"); // GroupNames label
G:=SmallGroup(240,124);
// by ID
G=gap.SmallGroup(240,124);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,964,5189,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^6=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;
// generators/relations
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