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## G = Q8.D15order 240 = 24·3·5

### The non-split extension by Q8 of D15 acting via D15/C5=S3

Aliases: Q8.D15, C10.1S4, C5⋊CSU2(𝔽3), SL2(𝔽3).D5, C2.2(C5⋊S4), (C5×Q8).1S3, (C5×SL2(𝔽3)).1C2, SmallGroup(240,105)

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=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

Character table of Q8.D15

 class 1 2 3 4A 4B 5A 5B 6 8A 8B 10A 10B 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D size 1 1 8 6 60 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 -1 2 0 2 2 -1 0 0 2 2 -1 -1 -1 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 2 2 0 -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 2 2 0 -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 -1 2 0 -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 ρ7 2 2 -1 2 0 -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 ρ8 2 2 -1 2 0 -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 ρ9 2 2 -1 2 0 -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 ρ10 2 -2 -1 0 0 2 2 1 √2 -√2 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ11 2 -2 -1 0 0 2 2 1 -√2 √2 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ12 3 3 0 -1 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 0 -1 -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 1 0 0 4 4 -1 0 0 -4 -4 1 1 1 1 0 0 -1 -1 -1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ15 4 -4 -2 0 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 symplectic faithful, Schur index 2 ρ16 4 -4 -2 0 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 symplectic faithful, Schur index 2 ρ17 4 -4 1 0 0 -1-√5 -1+√5 -1 0 0 1+√5 1-√5 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 0 0 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 symplectic faithful, Schur index 2 ρ18 4 -4 1 0 0 -1+√5 -1-√5 -1 0 0 1-√5 1+√5 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 0 0 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 symplectic faithful, Schur index 2 ρ19 4 -4 1 0 0 -1-√5 -1+√5 -1 0 0 1+√5 1-√5 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 0 0 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 symplectic faithful, Schur index 2 ρ20 4 -4 1 0 0 -1+√5 -1-√5 -1 0 0 1-√5 1+√5 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 0 0 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 symplectic faithful, Schur index 2 ρ21 6 6 0 -2 0 -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 -2 0 -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 80 points
Generators in S80
```(1 50 10 75)(2 41 6 66)(3 47 7 72)(4 38 8 78)(5 44 9 69)(11 24 17 60)(12 30 18 51)(13 21 19 57)(14 27 20 63)(15 33 16 54)(22 32 58 53)(23 64 59 28)(25 35 61 56)(26 52 62 31)(29 55 65 34)(36 46 76 71)(37 67 77 42)(39 49 79 74)(40 70 80 45)(43 73 68 48)
(1 40 10 80)(2 46 6 71)(3 37 7 77)(4 43 8 68)(5 49 9 74)(11 29 17 65)(12 35 18 56)(13 26 19 62)(14 32 20 53)(15 23 16 59)(21 31 57 52)(22 63 58 27)(24 34 60 55)(25 51 61 30)(28 54 64 33)(36 66 76 41)(38 48 78 73)(39 69 79 44)(42 72 67 47)(45 75 70 50)
(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 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 14 10 20)(2 13 6 19)(3 12 7 18)(4 11 8 17)(5 15 9 16)(21 36 57 76)(22 50 58 75)(23 49 59 74)(24 48 60 73)(25 47 61 72)(26 46 62 71)(27 45 63 70)(28 44 64 69)(29 43 65 68)(30 42 51 67)(31 41 52 66)(32 40 53 80)(33 39 54 79)(34 38 55 78)(35 37 56 77)```

`G:=sub<Sym(80)| (1,50,10,75)(2,41,6,66)(3,47,7,72)(4,38,8,78)(5,44,9,69)(11,24,17,60)(12,30,18,51)(13,21,19,57)(14,27,20,63)(15,33,16,54)(22,32,58,53)(23,64,59,28)(25,35,61,56)(26,52,62,31)(29,55,65,34)(36,46,76,71)(37,67,77,42)(39,49,79,74)(40,70,80,45)(43,73,68,48), (1,40,10,80)(2,46,6,71)(3,37,7,77)(4,43,8,68)(5,49,9,74)(11,29,17,65)(12,35,18,56)(13,26,19,62)(14,32,20,53)(15,23,16,59)(21,31,57,52)(22,63,58,27)(24,34,60,55)(25,51,61,30)(28,54,64,33)(36,66,76,41)(38,48,78,73)(39,69,79,44)(42,72,67,47)(45,75,70,50), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,14,10,20)(2,13,6,19)(3,12,7,18)(4,11,8,17)(5,15,9,16)(21,36,57,76)(22,50,58,75)(23,49,59,74)(24,48,60,73)(25,47,61,72)(26,46,62,71)(27,45,63,70)(28,44,64,69)(29,43,65,68)(30,42,51,67)(31,41,52,66)(32,40,53,80)(33,39,54,79)(34,38,55,78)(35,37,56,77)>;`

`G:=Group( (1,50,10,75)(2,41,6,66)(3,47,7,72)(4,38,8,78)(5,44,9,69)(11,24,17,60)(12,30,18,51)(13,21,19,57)(14,27,20,63)(15,33,16,54)(22,32,58,53)(23,64,59,28)(25,35,61,56)(26,52,62,31)(29,55,65,34)(36,46,76,71)(37,67,77,42)(39,49,79,74)(40,70,80,45)(43,73,68,48), (1,40,10,80)(2,46,6,71)(3,37,7,77)(4,43,8,68)(5,49,9,74)(11,29,17,65)(12,35,18,56)(13,26,19,62)(14,32,20,53)(15,23,16,59)(21,31,57,52)(22,63,58,27)(24,34,60,55)(25,51,61,30)(28,54,64,33)(36,66,76,41)(38,48,78,73)(39,69,79,44)(42,72,67,47)(45,75,70,50), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,14,10,20)(2,13,6,19)(3,12,7,18)(4,11,8,17)(5,15,9,16)(21,36,57,76)(22,50,58,75)(23,49,59,74)(24,48,60,73)(25,47,61,72)(26,46,62,71)(27,45,63,70)(28,44,64,69)(29,43,65,68)(30,42,51,67)(31,41,52,66)(32,40,53,80)(33,39,54,79)(34,38,55,78)(35,37,56,77) );`

`G=PermutationGroup([[(1,50,10,75),(2,41,6,66),(3,47,7,72),(4,38,8,78),(5,44,9,69),(11,24,17,60),(12,30,18,51),(13,21,19,57),(14,27,20,63),(15,33,16,54),(22,32,58,53),(23,64,59,28),(25,35,61,56),(26,52,62,31),(29,55,65,34),(36,46,76,71),(37,67,77,42),(39,49,79,74),(40,70,80,45),(43,73,68,48)], [(1,40,10,80),(2,46,6,71),(3,37,7,77),(4,43,8,68),(5,49,9,74),(11,29,17,65),(12,35,18,56),(13,26,19,62),(14,32,20,53),(15,23,16,59),(21,31,57,52),(22,63,58,27),(24,34,60,55),(25,51,61,30),(28,54,64,33),(36,66,76,41),(38,48,78,73),(39,69,79,44),(42,72,67,47),(45,75,70,50)], [(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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,14,10,20),(2,13,6,19),(3,12,7,18),(4,11,8,17),(5,15,9,16),(21,36,57,76),(22,50,58,75),(23,49,59,74),(24,48,60,73),(25,47,61,72),(26,46,62,71),(27,45,63,70),(28,44,64,69),(29,43,65,68),(30,42,51,67),(31,41,52,66),(32,40,53,80),(33,39,54,79),(34,38,55,78),(35,37,56,77)]])`

Q8.D15 is a maximal subgroup of   CSU2(𝔽3)⋊D5  Dic5.6S4  D5×CSU2(𝔽3)  D10.1S4  Q8.D30  C20.2S4  C20.6S4
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 30 158 0 0 127 211
,
 1 0 0 0 0 1 0 0 0 0 84 114 0 0 29 157
,
 175 113 0 0 128 62 0 0 0 0 240 240 0 0 1 0
,
 0 240 0 0 240 0 0 0 0 0 127 129 0 0 2 114
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,30,127,0,0,158,211],[1,0,0,0,0,1,0,0,0,0,84,29,0,0,114,157],[175,128,0,0,113,62,0,0,0,0,240,1,0,0,240,0],[0,240,0,0,240,0,0,0,0,0,127,2,0,0,129,114] >;`

Q8.D15 in GAP, Magma, Sage, TeX

`Q_8.D_{15}`
`% in TeX`

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

`G:=SmallGroup(240,105);`
`// by ID`

`G=gap.SmallGroup(240,105);`
`# by ID`

`G:=PCGroup([6,-2,-3,-5,-2,2,-2,720,49,434,1443,2169,117,904,1360,202,88]);`
`// Polycyclic`

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

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