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## G = D10.D6order 240 = 24·3·5

### 6th non-split extension by D10 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10.D6
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — D10.D6
 Lower central C15 — C30 — D10.D6
 Upper central C1 — C2 — C22

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

Smallest permutation representation of D10.D6
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  Dic102Dic3  D202Dic3  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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