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G = D10⋊D6order 240 = 24·3·5

4th semidirect product of D10 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10⋊D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — 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=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 [×6], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6, D6 [×6], C2×C6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10 [×6], C2×C10, C2×C10, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3 [×2], C5×S3, C3×D5, D15 [×2], D15, C30, C30, C4×D5, D20, C5⋊D4, C5⋊D4, C5×D4, C22×D5 [×2], S3×D4, C5×Dic3, C3×Dic5, S3×D5 [×2], C6×D5, S3×C10, D30 [×2], D30 [×2], C2×C30, D4×D5, D30.C2, C3⋊D20, C5⋊D12, C3×C5⋊D4, C5×C3⋊D4, C2×S3×D5, C22×D15, D10⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C22×D5, S3×D4, S3×D5, D4×D5, C2×S3×D5, D10⋊D6

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 15)(12 14)(16 20)(17 19)(21 30)(22 29)(23 28)(24 27)(25 26)(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 21 11)(2 41 36 54 22 12)(3 42 37 55 23 13)(4 43 38 56 24 14)(5 44 39 57 25 15)(6 45 40 58 26 16)(7 46 31 59 27 17)(8 47 32 60 28 18)(9 48 33 51 29 19)(10 49 34 52 30 20)
(1 11)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(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)(21,30)(22,29)(23,28)(24,27)(25,26)(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), (1,50,35,53,21,11)(2,41,36,54,22,12)(3,42,37,55,23,13)(4,43,38,56,24,14)(5,44,39,57,25,15)(6,45,40,58,26,16)(7,46,31,59,27,17)(8,47,32,60,28,18)(9,48,33,51,29,19)(10,49,34,52,30,20), (1,11)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(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)(21,30)(22,29)(23,28)(24,27)(25,26)(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), (1,50,35,53,21,11)(2,41,36,54,22,12)(3,42,37,55,23,13)(4,43,38,56,24,14)(5,44,39,57,25,15)(6,45,40,58,26,16)(7,46,31,59,27,17)(8,47,32,60,28,18)(9,48,33,51,29,19)(10,49,34,52,30,20), (1,11)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(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),(21,30),(22,29),(23,28),(24,27),(25,26),(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)], [(1,50,35,53,21,11),(2,41,36,54,22,12),(3,42,37,55,23,13),(4,43,38,56,24,14),(5,44,39,57,25,15),(6,45,40,58,26,16),(7,46,31,59,27,17),(8,47,32,60,28,18),(9,48,33,51,29,19),(10,49,34,52,30,20)], [(1,11),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,60),(39,59),(40,58)])`

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 S3×D4 S3×D5 D4×D5 C2×S3×D5 D10⋊D6 kernel D10⋊D6 D30.C2 C3⋊D20 C5⋊D12 C3×C5⋊D4 C5×C3⋊D4 C2×S3×D5 C22×D15 C5⋊D4 D15 C3⋊D4 Dic5 D10 C2×C10 Dic3 D6 C2×C6 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`

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