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## G = C42⋊D15order 480 = 25·3·5

### The semidirect product of C42 and D15 acting via D15/C5=S3

Aliases: C42⋊D15, C5⋊(C42⋊S3), (C4×C20)⋊1S3, C42⋊C32D5, C22.(C5⋊S4), (C2×C10).2S4, (C5×C42⋊C3)⋊4C2, SmallGroup(480,258)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C5×C42⋊C3 — C42⋊D15
 Chief series C1 — C22 — C42 — C4×C20 — C5×C42⋊C3 — C42⋊D15
 Lower central C5×C42⋊C3 — C42⋊D15
 Upper central C1

Generators and relations for C42⋊D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, cac-1=dbd=a-1b-1, ad=da, cbc-1=a, dcd=c-1 >

3C2
60C2
16C3
3C4
3C4
30C4
30C22
80S3
3C10
12D5
16C15
15Q8
15D4
30C8
30D4
30C2×C4
4A4
3C20
3C20
6D10
6Dic5
16D15
15M4(2)
15C4○D4
20S4
3D20
15C4≀C2

Character table of C42⋊D15

 class 1 2A 2B 3 4A 4B 4C 4D 5A 5B 8A 8B 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H size 1 3 60 32 3 3 6 60 2 2 60 60 6 6 32 32 32 32 6 6 6 6 6 6 6 6 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 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 linear of order 2 ρ3 2 2 0 -1 2 2 2 0 2 2 0 0 2 2 -1 -1 -1 -1 2 2 2 2 2 2 2 2 orthogonal lifted from S3 ρ4 2 2 0 2 2 2 2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ5 2 2 0 2 2 2 2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 0 -1 2 2 2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D15 ρ7 2 2 0 -1 2 2 2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D15 ρ8 2 2 0 -1 2 2 2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D15 ρ9 2 2 0 -1 2 2 2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D15 ρ10 3 3 1 0 -1 -1 -1 1 3 3 -1 -1 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ11 3 3 -1 0 -1 -1 -1 -1 3 3 1 1 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ12 3 -1 1 0 -1-2i -1+2i 1 -1 3 3 -i i -1 -1 0 0 0 0 1 -1-2i -1+2i 1 1 1 -1-2i -1+2i complex lifted from C42⋊S3 ρ13 3 -1 1 0 -1+2i -1-2i 1 -1 3 3 i -i -1 -1 0 0 0 0 1 -1+2i -1-2i 1 1 1 -1+2i -1-2i complex lifted from C42⋊S3 ρ14 3 -1 -1 0 -1+2i -1-2i 1 1 3 3 -i i -1 -1 0 0 0 0 1 -1+2i -1-2i 1 1 1 -1+2i -1-2i complex lifted from C42⋊S3 ρ15 3 -1 -1 0 -1-2i -1+2i 1 1 3 3 i -i -1 -1 0 0 0 0 1 -1-2i -1+2i 1 1 1 -1-2i -1+2i complex lifted from C42⋊S3 ρ16 6 -2 0 0 2 2 -2 0 6 6 0 0 -2 -2 0 0 0 0 -2 2 2 -2 -2 -2 2 2 orthogonal lifted from C42⋊S3 ρ17 6 6 0 0 -2 -2 -2 0 -3-3√5/2 -3+3√5/2 0 0 -3-3√5/2 -3+3√5/2 0 0 0 0 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from C5⋊S4 ρ18 6 6 0 0 -2 -2 -2 0 -3+3√5/2 -3-3√5/2 0 0 -3+3√5/2 -3-3√5/2 0 0 0 0 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from C5⋊S4 ρ19 6 -2 0 0 2 2 -2 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 -1+√5/2 -1-√5/2 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 -1-√5/2 -1+√5/2 orthogonal faithful ρ20 6 -2 0 0 2 2 -2 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 -1-√5/2 -1+√5/2 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 -1+√5/2 -1-√5/2 orthogonal faithful ρ21 6 -2 0 0 2 2 -2 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 -1-√5/2 -1+√5/2 -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 -1+√5/2 -1-√5/2 orthogonal faithful ρ22 6 -2 0 0 2 2 -2 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 -1+√5/2 -1-√5/2 -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 -1-√5/2 -1+√5/2 orthogonal faithful ρ23 6 -2 0 0 -2+4i -2-4i 2 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 -1-√5/2 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 -1-√5/2 -1+√5/2 -1+√5/2 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 complex faithful ρ24 6 -2 0 0 -2-4i -2+4i 2 0 -3-3√5/2 -3+3√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 -1-√5/2 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 -1-√5/2 -1+√5/2 -1+√5/2 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 complex faithful ρ25 6 -2 0 0 -2-4i -2+4i 2 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 -1+√5/2 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 -1+√5/2 -1-√5/2 -1-√5/2 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 complex faithful ρ26 6 -2 0 0 -2+4i -2-4i 2 0 -3+3√5/2 -3-3√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 -1+√5/2 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 -1+√5/2 -1-√5/2 -1-√5/2 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 complex faithful

Smallest permutation representation of C42⋊D15
On 60 points
Generators in S60
```(1 22 60 42)(2 46)(3 24 47 44)(4 25 48 45)(5 49)(6 27 50 32)(7 28 51 33)(8 52)(9 30 53 35)(10 16 54 36)(11 55)(12 18 56 38)(13 19 57 39)(14 58)(15 21 59 41)(17 37)(20 40)(23 43)(26 31)(29 34)
(1 22 60 42)(2 23 46 43)(3 47)(4 25 48 45)(5 26 49 31)(6 50)(7 28 51 33)(8 29 52 34)(9 53)(10 16 54 36)(11 17 55 37)(12 56)(13 19 57 39)(14 20 58 40)(15 59)(18 38)(21 41)(24 44)(27 32)(30 35)
(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 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 60)(16 32)(17 31)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)```

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

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

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

Matrix representation of C42⋊D15 in GL5(𝔽241)

 1 0 0 0 0 0 1 0 0 0 0 0 177 0 45 0 0 0 177 0 0 0 0 0 240
,
 1 0 0 0 0 0 1 0 0 0 0 0 177 107 0 0 0 0 240 0 0 0 0 0 177
,
 189 51 0 0 0 189 0 0 0 0 0 0 33 41 200 0 0 0 0 1 0 0 239 34 208
,
 1 240 0 0 0 0 240 0 0 0 0 0 34 25 216 0 0 2 207 33 0 0 0 0 240

`G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,1,0,0,0,0,0,177,0,0,0,0,0,177,0,0,0,45,0,240],[1,0,0,0,0,0,1,0,0,0,0,0,177,0,0,0,0,107,240,0,0,0,0,0,177],[189,189,0,0,0,51,0,0,0,0,0,0,33,0,239,0,0,41,0,34,0,0,200,1,208],[1,0,0,0,0,240,240,0,0,0,0,0,34,2,0,0,0,25,207,0,0,0,216,33,240] >;`

C42⋊D15 in GAP, Magma, Sage, TeX

`C_4^2\rtimes D_{15}`
`% in TeX`

`G:=Group("C4^2:D15");`
`// GroupNames label`

`G:=SmallGroup(480,258);`
`// by ID`

`G=gap.SmallGroup(480,258);`
`# by ID`

`G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,57,506,1683,850,360,1054,5786,102,15125,5052,8833]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^-1*b^-1,a*d=d*a,c*b*c^-1=a,d*c*d=c^-1>;`
`// generators/relations`

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