Aliases: Q8.1S4, C23.11S4, C22⋊CSU2(𝔽3), C2.2(C22⋊S4), Q8⋊A4.2C2, (C22×Q8).6S3, SmallGroup(192,1489)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊A4 — Q8.1S4 |
Generators and relations for Q8.1S4
G = < a,b,c,d,e,f | a4=c2=d2=e3=1, b2=f2=a2, bab-1=fbf-1=a-1, ac=ca, ad=da, eae-1=ab, faf-1=a2b, bc=cb, bd=db, ebe-1=a, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >
Subgroups: 293 in 64 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C8, C2×C4, Q8, Q8, C23, Dic3, A4, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C2×Q8, SL2(𝔽3), C2×A4, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, CSU2(𝔽3), A4⋊C4, C22⋊Q16, Q8⋊A4, Q8.1S4
Quotients: C1, C2, S3, S4, CSU2(𝔽3), C22⋊S4, Q8.1S4
Character table of Q8.1S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 3 | 3 | 32 | 6 | 6 | 12 | 24 | 24 | 32 | 12 | 12 | 12 | 12 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -√2 | √2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ5 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | √2 | -√2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ6 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | -1 | 1 | -1 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | -1 | 0 | -1 | 3 | -1 | -1 | 1 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | -1 | -1 | 0 | -1 | 3 | -1 | 1 | -1 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S4 |
| ρ10 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | -1 | -1 | 1 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | 3 | 3 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ12 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ13 | 6 | 6 | -2 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C22⋊S4 |
| ρ14 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic faithful, Schur index 2 |
| ρ15 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(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)
(1 10 3 12)(2 9 4 11)(5 19 7 17)(6 18 8 20)(13 43 15 41)(14 42 16 44)(21 36 23 34)(22 35 24 33)(25 31 27 29)(26 30 28 32)(37 48 39 46)(38 47 40 45)
(1 7)(2 8)(3 5)(4 6)(9 20)(10 17)(11 18)(12 19)(13 15)(14 16)(21 29)(22 30)(23 31)(24 32)(25 36)(26 33)(27 34)(28 35)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 38)(14 39)(15 40)(16 37)(17 19)(18 20)(21 31)(22 32)(23 29)(24 30)(25 34)(26 35)(27 36)(28 33)(41 45)(42 46)(43 47)(44 48)
(1 25 45)(2 31 37)(3 27 47)(4 29 39)(5 36 41)(6 23 16)(7 34 43)(8 21 14)(9 26 38)(10 30 46)(11 28 40)(12 32 48)(13 18 33)(15 20 35)(17 24 44)(19 22 42)
(1 5 3 7)(2 17 4 19)(6 10 8 12)(9 20 11 18)(13 26 15 28)(14 32 16 30)(21 48 23 46)(22 37 24 39)(25 41 27 43)(29 42 31 44)(33 38 35 40)(34 45 36 47)
G:=sub<Sym(48)| (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), (1,10,3,12)(2,9,4,11)(5,19,7,17)(6,18,8,20)(13,43,15,41)(14,42,16,44)(21,36,23,34)(22,35,24,33)(25,31,27,29)(26,30,28,32)(37,48,39,46)(38,47,40,45), (1,7)(2,8)(3,5)(4,6)(9,20)(10,17)(11,18)(12,19)(13,15)(14,16)(21,29)(22,30)(23,31)(24,32)(25,36)(26,33)(27,34)(28,35)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,38)(14,39)(15,40)(16,37)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30)(25,34)(26,35)(27,36)(28,33)(41,45)(42,46)(43,47)(44,48), (1,25,45)(2,31,37)(3,27,47)(4,29,39)(5,36,41)(6,23,16)(7,34,43)(8,21,14)(9,26,38)(10,30,46)(11,28,40)(12,32,48)(13,18,33)(15,20,35)(17,24,44)(19,22,42), (1,5,3,7)(2,17,4,19)(6,10,8,12)(9,20,11,18)(13,26,15,28)(14,32,16,30)(21,48,23,46)(22,37,24,39)(25,41,27,43)(29,42,31,44)(33,38,35,40)(34,45,36,47)>;
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), (1,10,3,12)(2,9,4,11)(5,19,7,17)(6,18,8,20)(13,43,15,41)(14,42,16,44)(21,36,23,34)(22,35,24,33)(25,31,27,29)(26,30,28,32)(37,48,39,46)(38,47,40,45), (1,7)(2,8)(3,5)(4,6)(9,20)(10,17)(11,18)(12,19)(13,15)(14,16)(21,29)(22,30)(23,31)(24,32)(25,36)(26,33)(27,34)(28,35)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,38)(14,39)(15,40)(16,37)(17,19)(18,20)(21,31)(22,32)(23,29)(24,30)(25,34)(26,35)(27,36)(28,33)(41,45)(42,46)(43,47)(44,48), (1,25,45)(2,31,37)(3,27,47)(4,29,39)(5,36,41)(6,23,16)(7,34,43)(8,21,14)(9,26,38)(10,30,46)(11,28,40)(12,32,48)(13,18,33)(15,20,35)(17,24,44)(19,22,42), (1,5,3,7)(2,17,4,19)(6,10,8,12)(9,20,11,18)(13,26,15,28)(14,32,16,30)(21,48,23,46)(22,37,24,39)(25,41,27,43)(29,42,31,44)(33,38,35,40)(34,45,36,47) );
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)], [(1,10,3,12),(2,9,4,11),(5,19,7,17),(6,18,8,20),(13,43,15,41),(14,42,16,44),(21,36,23,34),(22,35,24,33),(25,31,27,29),(26,30,28,32),(37,48,39,46),(38,47,40,45)], [(1,7),(2,8),(3,5),(4,6),(9,20),(10,17),(11,18),(12,19),(13,15),(14,16),(21,29),(22,30),(23,31),(24,32),(25,36),(26,33),(27,34),(28,35),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,38),(14,39),(15,40),(16,37),(17,19),(18,20),(21,31),(22,32),(23,29),(24,30),(25,34),(26,35),(27,36),(28,33),(41,45),(42,46),(43,47),(44,48)], [(1,25,45),(2,31,37),(3,27,47),(4,29,39),(5,36,41),(6,23,16),(7,34,43),(8,21,14),(9,26,38),(10,30,46),(11,28,40),(12,32,48),(13,18,33),(15,20,35),(17,24,44),(19,22,42)], [(1,5,3,7),(2,17,4,19),(6,10,8,12),(9,20,11,18),(13,26,15,28),(14,32,16,30),(21,48,23,46),(22,37,24,39),(25,41,27,43),(29,42,31,44),(33,38,35,40),(34,45,36,47)]])
Matrix representation of Q8.1S4 ►in GL5(𝔽73)
| 4 | 15 | 0 | 0 | 0 |
| 67 | 69 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 27 | 36 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 31 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 34 | 10 | 0 | 0 | 0 |
| 5 | 38 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 21 | 1 | 0 |
| 70 | 29 | 0 | 0 | 0 |
| 50 | 3 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 21 | 72 |
| 0 | 0 | 0 | 2 | 52 |
G:=sub<GL(5,GF(73))| [4,67,0,0,0,15,69,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[27,0,0,0,0,36,46,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,31,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[34,5,0,0,0,10,38,0,0,0,0,0,0,1,21,0,0,52,0,1,0,0,1,0,0],[70,50,0,0,0,29,3,0,0,0,0,0,72,0,0,0,0,0,21,2,0,0,0,72,52] >;
Q8.1S4 in GAP, Magma, Sage, TeX
Q_8._1S_4
% in TeX
G:=Group("Q8.1S4"); // GroupNames label
G:=SmallGroup(192,1489);
// by ID
G=gap.SmallGroup(192,1489);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,172,1013,2532,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^3=1,b^2=f^2=a^2,b*a*b^-1=f*b*f^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*b,f*a*f^-1=a^2*b,b*c=c*b,b*d=d*b,e*b*e^-1=a,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
Export