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
| C1 — C2 — Q8 — C5×SL2(𝔽3) — Q8⋊D15 |
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15 |
| C5×SL2(𝔽3) — Q8⋊D15 |
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 >
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 |
►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
Export
Subgroup lattice of Q8⋊D15 in TeX
Character table of Q8⋊D15 in TeX