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
| 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=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 |
►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
Export
Subgroup lattice of Q8.D15 in TeX
Character table of Q8.D15 in TeX