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## G = C24⋊2D15order 480 = 25·3·5

### 1st semidirect product of C24 and D15 acting via D15/C5=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×A4 — C24⋊2D15
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — C2×C5⋊S4 — C24⋊2D15
 Lower central C5×A4 — C10×A4 — C24⋊2D15
 Upper central C1 — C2 — C22

Generators and relations for C242D15
G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, faf=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fdf=cd=dc, cf=fc, ede-1=c, fef=e-1 >

Subgroups: 1036 in 124 conjugacy classes, 21 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, D10, C2×C10, C2×C10, C3⋊D4, S4, C2×A4, C2×A4, D15, C30, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, A4⋊C4, C2×S4, C22×A4, Dic15, C5×A4, D30, C2×C30, C23.D5, C2×C5⋊D4, C23×C10, A4⋊D4, C157D4, C5⋊S4, C10×A4, C10×A4, C242D5, A4⋊Dic5, C2×C5⋊S4, A4×C2×C10, C242D15
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3⋊D4, S4, D15, C5⋊D4, C2×S4, D30, A4⋊D4, C157D4, C5⋊S4, C2×C5⋊S4, C242D15

Smallest permutation representation of C242D15
On 60 points
Generators in S60
```(16 43)(17 44)(18 45)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 43)(17 44)(18 45)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
(2 57)(3 58)(5 60)(6 46)(8 48)(9 49)(11 51)(12 52)(14 54)(15 55)(17 44)(18 45)(20 32)(21 33)(23 35)(24 36)(26 38)(27 39)(29 41)(30 42)
(1 56)(3 58)(4 59)(6 46)(7 47)(9 49)(10 50)(12 52)(13 53)(15 55)(16 43)(18 45)(19 31)(21 33)(22 34)(24 36)(25 37)(27 39)(28 40)(30 42)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 50)(17 49)(18 48)(19 47)(20 46)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)```

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 3 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 2 3 3 6 60 8 60 60 60 2 2 8 8 8 2 ··· 2 6 ··· 6 8 8 8 8 8 ··· 8

46 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 6 6 6 6 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 S4 C2×S4 A4⋊D4 C5⋊S4 C2×C5⋊S4 C24⋊2D15 kernel C24⋊2D15 A4⋊Dic5 C2×C5⋊S4 A4×C2×C10 C23×C10 C5×A4 C22×A4 C22×C10 C2×A4 C2×C10 C24 A4 C23 C22 C2×C10 C10 C5 C22 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 4 4 4 8 2 2 1 2 2 4

Matrix representation of C242D15 in GL5(𝔽61)

 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 60 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 60 1 0 0 0 60 0 0 0 1 60 0
,
 39 19 0 0 0 19 39 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 22 19 0 0 0 42 39 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(61))| [0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[39,19,0,0,0,19,39,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[22,42,0,0,0,19,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C242D15 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_2D_{15}`
`% in TeX`

`G:=Group("C2^4:2D15");`
`// GroupNames label`

`G:=SmallGroup(480,1034);`
`// by ID`

`G=gap.SmallGroup(480,1034);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,451,3364,10085,1286,5886,2232]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*d*f=c*d=d*c,c*f=f*c,e*d*e^-1=c,f*e*f=e^-1>;`
`// generators/relations`

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