Aliases: Q8.2Dic6, C4.CSU2(𝔽3), SL2(𝔽3)⋊2Q8, (C2×C4).10S4, (C4×Q8).7S3, (C2×Q8).7D6, C2.5(A4⋊Q8), Q8⋊Dic3.2C2, C22.35(C2×S4), C2.3(C4.3S4), (C4×SL2(𝔽3)).6C2, C2.4(C2×CSU2(𝔽3)), (C2×SL2(𝔽3)).7C22, SmallGroup(192,950)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×SL2(𝔽3) — SL2(𝔽3)⋊Q8 |
| SL2(𝔽3) — C2×SL2(𝔽3) — SL2(𝔽3)⋊Q8 |
Generators and relations for SL2(𝔽3)⋊Q8
G = < a,b,c,d,e | a4=c3=d4=1, b2=a2, e2=d2, bab-1=dad-1=ebe-1=a-1, cac-1=b, eae-1=dbd-1=a2b, cbc-1=ab, dcd-1=a2bc, ece-1=c-1, ede-1=d-1 >
Subgroups: 221 in 59 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C4⋊Dic3, C2×SL2(𝔽3), C4.Q16, Q8⋊Dic3, C4×SL2(𝔽3), SL2(𝔽3)⋊Q8
Quotients: C1, C2, C22, S3, Q8, D6, Dic6, S4, CSU2(𝔽3), C2×S4, A4⋊Q8, C2×CSU2(𝔽3), C4.3S4, SL2(𝔽3)⋊Q8
Character table of SL2(𝔽3)⋊Q8
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 6 | 12 | 24 | 24 | 8 | 8 | 8 | 12 | 12 | 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 | 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 | linear of order 2 |
| ρ3 | 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 |
| ρ4 | 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 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ8 | 2 | -2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | √2 | √2 | -√2 | -1 | 1 | -1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√2 | -√2 | √2 | √2 | 1 | -1 | 1 | -1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | -2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √2 | -√2 | -√2 | √2 | -1 | 1 | -1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √2 | √2 | -√2 | -√2 | 1 | -1 | 1 | -1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ12 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ19 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ21 | 4 | -4 | 4 | -4 | 1 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ22 | 4 | -4 | 4 | -4 | 1 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ23 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
►On 64 points
Generators in S64
(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)
(1 5 3 7)(2 8 4 6)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)(33 40 35 38)(34 39 36 37)(41 48 43 46)(42 47 44 45)(49 56 51 54)(50 55 52 53)(57 64 59 62)(58 63 60 61)
(2 8 5)(4 6 7)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)(33 40 37)(35 38 39)(41 48 45)(43 46 47)(49 56 53)(51 54 55)(57 64 61)(59 62 63)
(1 32 9 22)(2 31 10 21)(3 30 11 24)(4 29 12 23)(5 28 15 18)(6 27 16 17)(7 26 13 20)(8 25 14 19)(33 55 43 61)(34 54 44 64)(35 53 41 63)(36 56 42 62)(37 49 47 59)(38 52 48 58)(39 51 45 57)(40 50 46 60)
(1 44 9 34)(2 47 10 37)(3 42 11 36)(4 45 12 39)(5 43 15 33)(6 48 16 38)(7 41 13 35)(8 46 14 40)(17 58 27 52)(18 61 28 55)(19 60 25 50)(20 63 26 53)(21 59 31 49)(22 64 32 54)(23 57 29 51)(24 62 30 56)
G:=sub<Sym(64)| (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), (1,5,3,7)(2,8,4,6)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,40,35,38)(34,39,36,37)(41,48,43,46)(42,47,44,45)(49,56,51,54)(50,55,52,53)(57,64,59,62)(58,63,60,61), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29)(33,40,37)(35,38,39)(41,48,45)(43,46,47)(49,56,53)(51,54,55)(57,64,61)(59,62,63), (1,32,9,22)(2,31,10,21)(3,30,11,24)(4,29,12,23)(5,28,15,18)(6,27,16,17)(7,26,13,20)(8,25,14,19)(33,55,43,61)(34,54,44,64)(35,53,41,63)(36,56,42,62)(37,49,47,59)(38,52,48,58)(39,51,45,57)(40,50,46,60), (1,44,9,34)(2,47,10,37)(3,42,11,36)(4,45,12,39)(5,43,15,33)(6,48,16,38)(7,41,13,35)(8,46,14,40)(17,58,27,52)(18,61,28,55)(19,60,25,50)(20,63,26,53)(21,59,31,49)(22,64,32,54)(23,57,29,51)(24,62,30,56)>;
G:=Group( (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), (1,5,3,7)(2,8,4,6)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,40,35,38)(34,39,36,37)(41,48,43,46)(42,47,44,45)(49,56,51,54)(50,55,52,53)(57,64,59,62)(58,63,60,61), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29)(33,40,37)(35,38,39)(41,48,45)(43,46,47)(49,56,53)(51,54,55)(57,64,61)(59,62,63), (1,32,9,22)(2,31,10,21)(3,30,11,24)(4,29,12,23)(5,28,15,18)(6,27,16,17)(7,26,13,20)(8,25,14,19)(33,55,43,61)(34,54,44,64)(35,53,41,63)(36,56,42,62)(37,49,47,59)(38,52,48,58)(39,51,45,57)(40,50,46,60), (1,44,9,34)(2,47,10,37)(3,42,11,36)(4,45,12,39)(5,43,15,33)(6,48,16,38)(7,41,13,35)(8,46,14,40)(17,58,27,52)(18,61,28,55)(19,60,25,50)(20,63,26,53)(21,59,31,49)(22,64,32,54)(23,57,29,51)(24,62,30,56) );
G=PermutationGroup([[(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)], [(1,5,3,7),(2,8,4,6),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32),(33,40,35,38),(34,39,36,37),(41,48,43,46),(42,47,44,45),(49,56,51,54),(50,55,52,53),(57,64,59,62),(58,63,60,61)], [(2,8,5),(4,6,7),(10,14,15),(12,16,13),(18,22,23),(20,24,21),(26,30,31),(28,32,29),(33,40,37),(35,38,39),(41,48,45),(43,46,47),(49,56,53),(51,54,55),(57,64,61),(59,62,63)], [(1,32,9,22),(2,31,10,21),(3,30,11,24),(4,29,12,23),(5,28,15,18),(6,27,16,17),(7,26,13,20),(8,25,14,19),(33,55,43,61),(34,54,44,64),(35,53,41,63),(36,56,42,62),(37,49,47,59),(38,52,48,58),(39,51,45,57),(40,50,46,60)], [(1,44,9,34),(2,47,10,37),(3,42,11,36),(4,45,12,39),(5,43,15,33),(6,48,16,38),(7,41,13,35),(8,46,14,40),(17,58,27,52),(18,61,28,55),(19,60,25,50),(20,63,26,53),(21,59,31,49),(22,64,32,54),(23,57,29,51),(24,62,30,56)]])
Matrix representation of SL2(𝔽3)⋊Q8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 72 |
| 0 | 0 | 9 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 0 | 65 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 66 | 14 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 0 | 8 | 65 |
| 0 | 0 | 72 | 65 |
| 61 | 22 | 0 | 0 |
| 10 | 12 | 0 | 0 |
| 0 | 0 | 2 | 18 |
| 0 | 0 | 20 | 71 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,9,9,0,0,72,64],[1,0,0,0,0,1,0,0,0,0,0,65,0,0,64,0],[0,1,0,0,72,72,0,0,0,0,72,72,0,0,1,0],[66,59,0,0,14,7,0,0,0,0,8,72,0,0,65,65],[61,10,0,0,22,12,0,0,0,0,2,20,0,0,18,71] >;
SL2(𝔽3)⋊Q8 in GAP, Magma, Sage, TeX
{\rm SL}_2({\mathbb F}_3)\rtimes Q_8 % in TeX
G:=Group("SL(2,3):Q8"); // GroupNames label
G:=SmallGroup(192,950);
// by ID
G=gap.SmallGroup(192,950);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,700,85,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^4=1,b^2=a^2,e^2=d^2,b*a*b^-1=d*a*d^-1=e*b*e^-1=a^-1,c*a*c^-1=b,e*a*e^-1=d*b*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=a^2*b*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
Export