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## G = D10⋊S4order 480 = 25·3·5

### 1st semidirect product of D10 and S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×A4 — D10⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — C2×D5×A4 — D10⋊S4
 Lower central C5×A4 — C10×A4 — D10⋊S4
 Upper central C1 — C2

Generators and relations for D10⋊S4
G = < a,b,c,d,e,f | a10=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a5b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 1012 in 124 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, A4, D6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C3⋊D4, S4, C2×A4, C2×A4, C5×S3, C3×D5, C30, C22≀C2, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, A4⋊C4, C2×S4, C22×A4, Dic15, C5×A4, C6×D5, S3×C10, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, A4⋊D4, C15⋊D4, C5×S4, D5×A4, C10×A4, C23⋊D10, A4⋊Dic5, C10×S4, C2×D5×A4, D10⋊S4
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3⋊D4, S4, C5⋊D4, C2×S4, S3×D5, A4⋊D4, C15⋊D4, D5×S4, D10⋊S4

Smallest permutation representation of D10⋊S4
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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 60)(30 59)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 13 23)(2 14 24)(3 15 25)(4 16 26)(5 17 27)(6 18 28)(7 19 29)(8 20 30)(9 11 21)(10 12 22)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)
(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 15A 15B 20A 20B 20C 20D 30A 30B order 1 2 2 2 2 2 2 3 4 4 4 5 5 6 6 6 10 10 10 10 10 10 10 10 10 10 15 15 20 20 20 20 30 30 size 1 1 3 3 10 12 30 8 12 60 60 2 2 8 40 40 2 2 6 6 6 6 12 12 12 12 16 16 12 12 12 12 16 16

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 3 3 4 4 6 6 6 type + + + + + + + + + + + + - + + image C1 C2 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 C5⋊D4 S4 C2×S4 S3×D5 C15⋊D4 A4⋊D4 D5×S4 D10⋊S4 kernel D10⋊S4 A4⋊Dic5 C10×S4 C2×D5×A4 C23×D5 C5×A4 C2×S4 C22×C10 C2×A4 C2×C10 A4 D10 C10 C23 C22 C5 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 4 2 2 2 2 1 4 4

Matrix representation of D10⋊S4 in GL5(𝔽61)

 3 0 0 0 0 23 41 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 54 0 0 0 33 54 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 60 0 1 0 0 60 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 60 0 0 1 0 60 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 60 0 0 0 0 59 1 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 60

`G:=sub<GL(5,GF(61))| [3,23,0,0,0,0,41,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,33,0,0,0,54,54,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[60,59,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60] >;`

D10⋊S4 in GAP, Magma, Sage, TeX

`D_{10}\rtimes S_4`
`% in TeX`

`G:=Group("D10:S4");`
`// GroupNames label`

`G:=SmallGroup(480,980);`
`// by ID`

`G=gap.SmallGroup(480,980);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,234,3364,5052,1286,2953,2232]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^5*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;`
`// generators/relations`

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