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G = Q8⋊D15order 240 = 24·3·5

The semidirect product of Q8 and D15 acting via D15/C5=S3

Aliases: Q8⋊D15, C10.2S4, C5⋊GL2(𝔽3), SL2(𝔽3)⋊D5, C2.3(C5⋊S4), (C5×Q8)⋊1S3, (C5×SL2(𝔽3))⋊1C2, SmallGroup(240,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — Q8⋊D15
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15
 Lower central C5×SL2(𝔽3) — Q8⋊D15
 Upper central C1 — C2

Generators and relations for Q8⋊D15
G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=a-1, cac-1=b, dad=a-1b, cbc-1=ab, dbd=a2b, dcd=c-1 >

60C2
4C3
3C4
30C22
4C6
20S3
20S3
12D5
4C15
15D4
15C8
20D6
3C20
6D10
4D15
4D15
4C30
15SD16
3D20
4D30

Character table of Q8⋊D15

 class 1 2A 2B 3 4 5A 5B 6 8A 8B 10A 10B 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D size 1 1 60 8 6 2 2 8 30 30 2 2 8 8 8 8 12 12 8 8 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 trivial ρ2 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 -1 0 0 2 2 -1 -1 -1 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 0 2 2 -1+√5/2 -1-√5/2 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 orthogonal lifted from D5 ρ5 2 2 0 2 2 -1-√5/2 -1+√5/2 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 orthogonal lifted from D5 ρ6 2 2 0 -1 2 -1+√5/2 -1-√5/2 -1 0 0 -1+√5/2 -1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 -1+√5/2 -1-√5/2 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal lifted from D15 ρ7 2 2 0 -1 2 -1-√5/2 -1+√5/2 -1 0 0 -1-√5/2 -1+√5/2 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -1-√5/2 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 orthogonal lifted from D15 ρ8 2 2 0 -1 2 -1-√5/2 -1+√5/2 -1 0 0 -1-√5/2 -1+√5/2 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -1-√5/2 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 orthogonal lifted from D15 ρ9 2 2 0 -1 2 -1+√5/2 -1-√5/2 -1 0 0 -1+√5/2 -1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -1+√5/2 -1-√5/2 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 orthogonal lifted from D15 ρ10 2 -2 0 -1 0 2 2 1 -√-2 √-2 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 complex lifted from GL2(𝔽3) ρ11 2 -2 0 -1 0 2 2 1 √-2 -√-2 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 complex lifted from GL2(𝔽3) ρ12 3 3 -1 0 -1 3 3 0 1 1 3 3 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ13 3 3 1 0 -1 3 3 0 -1 -1 3 3 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ14 4 -4 0 1 0 4 4 -1 0 0 -4 -4 1 1 1 1 0 0 -1 -1 -1 -1 orthogonal lifted from GL2(𝔽3) ρ15 4 -4 0 -2 0 -1+√5 -1-√5 2 0 0 1-√5 1+√5 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal faithful, Schur index 2 ρ16 4 -4 0 -2 0 -1-√5 -1+√5 2 0 0 1+√5 1-√5 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal faithful, Schur index 2 ρ17 4 -4 0 1 0 -1+√5 -1-√5 -1 0 0 1-√5 1+√5 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 -ζ3ζ54+ζ3ζ5+ζ5 -ζ32ζ54+ζ32ζ5+ζ5 0 0 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal faithful ρ18 4 -4 0 1 0 -1-√5 -1+√5 -1 0 0 1+√5 1-√5 -ζ32ζ54+ζ32ζ5+ζ5 -ζ3ζ54+ζ3ζ5+ζ5 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 0 0 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 orthogonal faithful ρ19 4 -4 0 1 0 -1+√5 -1-√5 -1 0 0 1-√5 1+√5 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 -ζ32ζ54+ζ32ζ5+ζ5 -ζ3ζ54+ζ3ζ5+ζ5 0 0 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 orthogonal faithful ρ20 4 -4 0 1 0 -1-√5 -1+√5 -1 0 0 1+√5 1-√5 -ζ3ζ54+ζ3ζ5+ζ5 -ζ32ζ54+ζ32ζ5+ζ5 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 0 0 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 orthogonal faithful ρ21 6 6 0 0 -2 -3-3√5/2 -3+3√5/2 0 0 0 -3-3√5/2 -3+3√5/2 0 0 0 0 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from C5⋊S4 ρ22 6 6 0 0 -2 -3+3√5/2 -3-3√5/2 0 0 0 -3+3√5/2 -3-3√5/2 0 0 0 0 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from C5⋊S4

Smallest permutation representation of Q8⋊D15
On 40 points
Generators in S40
```(1 39 6 23)(2 30 7 14)(3 36 8 20)(4 27 9 11)(5 33 10 17)(12 22 28 38)(13 34 29 18)(15 25 31 26)(16 37 32 21)(19 40 35 24)
(1 29 6 13)(2 35 7 19)(3 26 8 25)(4 32 9 16)(5 38 10 22)(11 21 27 37)(12 33 28 17)(14 24 30 40)(15 36 31 20)(18 39 34 23)
(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)
(1 2)(3 5)(6 7)(8 10)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 40)(24 39)(25 38)```

`G:=sub<Sym(40)| (1,39,6,23)(2,30,7,14)(3,36,8,20)(4,27,9,11)(5,33,10,17)(12,22,28,38)(13,34,29,18)(15,25,31,26)(16,37,32,21)(19,40,35,24), (1,29,6,13)(2,35,7,19)(3,26,8,25)(4,32,9,16)(5,38,10,22)(11,21,27,37)(12,33,28,17)(14,24,30,40)(15,36,31,20)(18,39,34,23), (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), (1,2)(3,5)(6,7)(8,10)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,40)(24,39)(25,38)>;`

`G:=Group( (1,39,6,23)(2,30,7,14)(3,36,8,20)(4,27,9,11)(5,33,10,17)(12,22,28,38)(13,34,29,18)(15,25,31,26)(16,37,32,21)(19,40,35,24), (1,29,6,13)(2,35,7,19)(3,26,8,25)(4,32,9,16)(5,38,10,22)(11,21,27,37)(12,33,28,17)(14,24,30,40)(15,36,31,20)(18,39,34,23), (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), (1,2)(3,5)(6,7)(8,10)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,40)(24,39)(25,38) );`

`G=PermutationGroup([[(1,39,6,23),(2,30,7,14),(3,36,8,20),(4,27,9,11),(5,33,10,17),(12,22,28,38),(13,34,29,18),(15,25,31,26),(16,37,32,21),(19,40,35,24)], [(1,29,6,13),(2,35,7,19),(3,26,8,25),(4,32,9,16),(5,38,10,22),(11,21,27,37),(12,33,28,17),(14,24,30,40),(15,36,31,20),(18,39,34,23)], [(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)], [(1,2),(3,5),(6,7),(8,10),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,40),(24,39),(25,38)]])`

Q8⋊D15 is a maximal subgroup of   Dic5.7S4  GL2(𝔽3)⋊D5  D10.2S4  D5×GL2(𝔽3)  Q8.D30  C20.6S4  C20.3S4
Q8⋊D15 is a maximal quotient of   Q8⋊Dic15

Matrix representation of Q8⋊D15 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 157 114 0 0 29 84
,
 1 0 0 0 0 1 0 0 0 0 113 212 0 0 83 128
,
 66 210 0 0 124 128 0 0 0 0 211 157 0 0 128 29
,
 8 230 0 0 181 233 0 0 0 0 238 134 0 0 169 3
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,157,29,0,0,114,84],[1,0,0,0,0,1,0,0,0,0,113,83,0,0,212,128],[66,124,0,0,210,128,0,0,0,0,211,128,0,0,157,29],[8,181,0,0,230,233,0,0,0,0,238,169,0,0,134,3] >;`

Q8⋊D15 in GAP, Magma, Sage, TeX

`Q_8\rtimes D_{15}`
`% in TeX`

`G:=Group("Q8:D15");`
`// GroupNames label`

`G:=SmallGroup(240,106);`
`// by ID`

`G=gap.SmallGroup(240,106);`
`# by ID`

`G:=PCGroup([6,-2,-3,-5,-2,2,-2,49,434,1443,2169,117,904,1360,202,88]);`
`// Polycyclic`

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

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