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G = D6⋊F5order 240 = 24·3·5

The semidirect product of D6 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D6⋊F5
G = < a,b,c,d | a6=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >

Subgroups: 392 in 68 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C22 [×5], C5, S3 [×2], C6, C6 [×2], C2×C4 [×2], C23, D5 [×2], D5, C10, C10, Dic3, C12, D6, D6 [×3], C2×C6, C15, C22⋊C4, F5 [×2], D10, D10 [×3], C2×C10, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5 [×2], D15, C30, C2×F5, C2×F5, C22×D5, D6⋊C4, C3×F5, C3⋊F5, S3×D5 [×2], C6×D5, S3×C10, D30, C22⋊F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D6⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C2×F5, D6⋊C4, C22⋊F5, S3×F5, D6⋊F5

Character table of D6⋊F5

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5 6A 6B 6C 10A 10B 10C 12A 12B 12C 12D 15 30 size 1 1 5 5 6 30 2 10 10 30 30 4 2 10 10 4 12 12 10 10 10 10 8 8 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 -1 1 1 -i i i -i 1 1 -1 -1 1 -1 -1 i i -i -i 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 -i -i i i 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 -i -i i i 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 i i -i -i 1 1 linear of order 4 ρ9 2 2 2 2 0 0 -1 -2 -2 0 0 2 -1 -1 -1 2 0 0 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 -1 2 2 0 0 2 -1 -1 -1 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 -2 2 0 0 2 0 0 0 0 2 -2 2 -2 -2 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ12 2 -2 2 -2 0 0 2 0 0 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ13 2 -2 -2 2 0 0 -1 0 0 0 0 2 1 -1 1 -2 0 0 √3 -√3 √3 -√3 -1 1 orthogonal lifted from D12 ρ14 2 -2 -2 2 0 0 -1 0 0 0 0 2 1 -1 1 -2 0 0 -√3 √3 -√3 √3 -1 1 orthogonal lifted from D12 ρ15 2 2 -2 -2 0 0 -1 -2i 2i 0 0 2 -1 1 1 2 0 0 -i -i i i -1 -1 complex lifted from C4×S3 ρ16 2 2 -2 -2 0 0 -1 2i -2i 0 0 2 -1 1 1 2 0 0 i i -i -i -1 -1 complex lifted from C4×S3 ρ17 2 -2 2 -2 0 0 -1 0 0 0 0 2 1 1 -1 -2 0 0 √-3 -√-3 -√-3 √-3 -1 1 complex lifted from C3⋊D4 ρ18 2 -2 2 -2 0 0 -1 0 0 0 0 2 1 1 -1 -2 0 0 -√-3 √-3 √-3 -√-3 -1 1 complex lifted from C3⋊D4 ρ19 4 4 0 0 4 0 4 0 0 0 0 -1 4 0 0 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from F5 ρ20 4 4 0 0 -4 0 4 0 0 0 0 -1 4 0 0 -1 1 1 0 0 0 0 -1 -1 orthogonal lifted from C2×F5 ρ21 4 -4 0 0 0 0 4 0 0 0 0 -1 -4 0 0 1 -√5 √5 0 0 0 0 -1 1 orthogonal lifted from C22⋊F5 ρ22 4 -4 0 0 0 0 4 0 0 0 0 -1 -4 0 0 1 √5 -√5 0 0 0 0 -1 1 orthogonal lifted from C22⋊F5 ρ23 8 8 0 0 0 0 -4 0 0 0 0 -2 -4 0 0 -2 0 0 0 0 0 0 1 1 orthogonal lifted from S3×F5 ρ24 8 -8 0 0 0 0 -4 0 0 0 0 -2 4 0 0 2 0 0 0 0 0 0 1 -1 orthogonal faithful, Schur index 2

Smallest permutation representation of D6⋊F5
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 6)(2 5)(3 4)(7 9)(10 12)(13 15)(16 18)(19 21)(22 24)(26 30)(27 29)(31 34)(32 33)(35 36)(37 38)(39 42)(40 41)(43 44)(45 48)(46 47)(49 50)(51 54)(52 53)(55 59)(56 58)
(1 38 53 44 36)(2 39 54 45 31)(3 40 49 46 32)(4 41 50 47 33)(5 42 51 48 34)(6 37 52 43 35)(7 59 16 19 27)(8 60 17 20 28)(9 55 18 21 29)(10 56 13 22 30)(11 57 14 23 25)(12 58 15 24 26)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 47 16 50)(8 48 17 51)(9 43 18 52)(10 44 13 53)(11 45 14 54)(12 46 15 49)(19 33 27 41)(20 34 28 42)(21 35 29 37)(22 36 30 38)(23 31 25 39)(24 32 26 40)```

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

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

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

D6⋊F5 is a maximal subgroup of
C4⋊F53S3  (C4×S3)⋊F5  F5×D12  D603C4  F5×C3⋊D4  S3×C22⋊F5  C3⋊D4⋊F5
D6⋊F5 is a maximal quotient of
D60⋊C4  D12⋊F5  Dic6⋊F5  Dic30⋊C4  D124F5  D122F5  D602C4  D605C4  D10.20D12  Dic5.22D12  D30⋊C8  D10.D12  D10.4D12  Dic5.D12  Dic5.4D12

Matrix representation of D6⋊F5 in GL6(𝔽61)

 2 46 0 0 0 0 49 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 46 0 0 0 0 49 59 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 45 44 0 0 0 0 0 0 17 18 0 0 0 0 44 0
,
 53 19 0 0 0 0 3 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 17 18 0 0 0 0 45 44 0 0

`G:=sub<GL(6,GF(61))| [2,49,0,0,0,0,46,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,49,0,0,0,0,46,59,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,45,0,0,0,0,60,44,0,0,0,0,0,0,17,44,0,0,0,0,18,0],[53,3,0,0,0,0,19,8,0,0,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,1,0,0,0,0,0,0,1,0,0] >;`

D6⋊F5 in GAP, Magma, Sage, TeX

`D_6\rtimes F_5`
`% in TeX`

`G:=Group("D6:F5");`
`// GroupNames label`

`G:=SmallGroup(240,96);`
`// by ID`

`G=gap.SmallGroup(240,96);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,490,3461,1745]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;`
`// generators/relations`

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