Aliases: Q8.4S4, U2(𝔽3)⋊1C2, SL2(𝔽3).8D4, 2+ 1+4.1S3, C4.5(C2×S4), Q8.A4.C2, C4○D4.1D6, C4.S4⋊2C2, Q8.4(C3⋊D4), C4.A4.1C22, C2.12(A4⋊D4), SmallGroup(192,987)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.4S4
G = < a,b,c,d,e,f | a4=e3=1, b2=c2=d2=f2=a2, bab-1=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=a-1b, dcd-1=a2c, ece-1=a2cd, fcf-1=cd, ede-1=c, fdf-1=a2d, fef-1=e-1 >
Subgroups: 293 in 69 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), Dic6, C3×Q8, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C3⋊Q16, CSU2(𝔽3), C4.A4, C4.A4, D4.9D4, U2(𝔽3), C4.S4, Q8.A4, Q8.4S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, Q8.4S4
Character table of Q8.4S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 12A | 12B | 12C | |
| size | 1 | 1 | 6 | 12 | 8 | 2 | 4 | 6 | 12 | 12 | 24 | 8 | 24 | 24 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | -2 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | √-3 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | -√-3 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ10 | 3 | 3 | -1 | 1 | 0 | 3 | -3 | -1 | 1 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ11 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | -1 | 0 | 1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -1 | 1 | 0 | 3 | -3 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 6 | 6 | 2 | 0 | 0 | -6 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
| ρ17 | 8 | -8 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 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 9 3 11)(2 12 4 10)(5 46 7 48)(6 45 8 47)(13 18 15 20)(14 17 16 19)(21 27 23 25)(22 26 24 28)(29 34 31 36)(30 33 32 35)(37 42 39 44)(38 41 40 43)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 19 15 17)(14 20 16 18)(21 25 23 27)(22 26 24 28)(29 36 31 34)(30 33 32 35)(37 43 39 41)(38 44 40 42)(45 46 47 48)
(1 9 3 11)(2 10 4 12)(5 46 7 48)(6 47 8 45)(13 14 15 16)(17 20 19 18)(21 28 23 26)(22 25 24 27)(29 35 31 33)(30 36 32 34)(37 38 39 40)(41 44 43 42)
(1 14 22)(2 15 23)(3 16 24)(4 13 21)(5 43 35)(6 44 36)(7 41 33)(8 42 34)(9 17 26)(10 18 27)(11 19 28)(12 20 25)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 32 3 30)(2 31 4 29)(5 25 7 27)(6 28 8 26)(9 36 11 34)(10 35 12 33)(13 37 15 39)(14 40 16 38)(17 44 19 42)(18 43 20 41)(21 45 23 47)(22 48 24 46)
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), (1,9,3,11)(2,12,4,10)(5,46,7,48)(6,45,8,47)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,19,15,17)(14,20,16,18)(21,25,23,27)(22,26,24,28)(29,36,31,34)(30,33,32,35)(37,43,39,41)(38,44,40,42)(45,46,47,48), (1,9,3,11)(2,10,4,12)(5,46,7,48)(6,47,8,45)(13,14,15,16)(17,20,19,18)(21,28,23,26)(22,25,24,27)(29,35,31,33)(30,36,32,34)(37,38,39,40)(41,44,43,42), (1,14,22)(2,15,23)(3,16,24)(4,13,21)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,17,26)(10,18,27)(11,19,28)(12,20,25)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,32,3,30)(2,31,4,29)(5,25,7,27)(6,28,8,26)(9,36,11,34)(10,35,12,33)(13,37,15,39)(14,40,16,38)(17,44,19,42)(18,43,20,41)(21,45,23,47)(22,48,24,46)>;
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), (1,9,3,11)(2,12,4,10)(5,46,7,48)(6,45,8,47)(13,18,15,20)(14,17,16,19)(21,27,23,25)(22,26,24,28)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,19,15,17)(14,20,16,18)(21,25,23,27)(22,26,24,28)(29,36,31,34)(30,33,32,35)(37,43,39,41)(38,44,40,42)(45,46,47,48), (1,9,3,11)(2,10,4,12)(5,46,7,48)(6,47,8,45)(13,14,15,16)(17,20,19,18)(21,28,23,26)(22,25,24,27)(29,35,31,33)(30,36,32,34)(37,38,39,40)(41,44,43,42), (1,14,22)(2,15,23)(3,16,24)(4,13,21)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,17,26)(10,18,27)(11,19,28)(12,20,25)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,32,3,30)(2,31,4,29)(5,25,7,27)(6,28,8,26)(9,36,11,34)(10,35,12,33)(13,37,15,39)(14,40,16,38)(17,44,19,42)(18,43,20,41)(21,45,23,47)(22,48,24,46) );
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)], [(1,9,3,11),(2,12,4,10),(5,46,7,48),(6,45,8,47),(13,18,15,20),(14,17,16,19),(21,27,23,25),(22,26,24,28),(29,34,31,36),(30,33,32,35),(37,42,39,44),(38,41,40,43)], [(1,2,3,4),(5,8,7,6),(9,12,11,10),(13,19,15,17),(14,20,16,18),(21,25,23,27),(22,26,24,28),(29,36,31,34),(30,33,32,35),(37,43,39,41),(38,44,40,42),(45,46,47,48)], [(1,9,3,11),(2,10,4,12),(5,46,7,48),(6,47,8,45),(13,14,15,16),(17,20,19,18),(21,28,23,26),(22,25,24,27),(29,35,31,33),(30,36,32,34),(37,38,39,40),(41,44,43,42)], [(1,14,22),(2,15,23),(3,16,24),(4,13,21),(5,43,35),(6,44,36),(7,41,33),(8,42,34),(9,17,26),(10,18,27),(11,19,28),(12,20,25),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(1,32,3,30),(2,31,4,29),(5,25,7,27),(6,28,8,26),(9,36,11,34),(10,35,12,33),(13,37,15,39),(14,40,16,38),(17,44,19,42),(18,43,20,41),(21,45,23,47),(22,48,24,46)]])
Matrix representation of Q8.4S4 ►in GL4(𝔽5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 3 |
| 0 | 1 | 2 | 0 |
| 0 | 4 | 4 | 0 |
| 1 | 0 | 0 | 4 |
| 2 | 0 | 0 | 1 |
| 0 | 2 | 4 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 3 |
| 0 | 2 | 2 | 0 |
| 0 | 4 | 2 | 0 |
| 1 | 0 | 0 | 2 |
| 0 | 4 | 2 | 0 |
| 2 | 0 | 0 | 4 |
| 3 | 0 | 0 | 2 |
| 0 | 4 | 4 | 0 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,2,0,0,2,0,0,0,0,0,0,3,0,0,3,0],[1,0,0,1,0,1,4,0,0,2,4,0,3,0,0,4],[2,0,0,0,0,2,0,0,0,4,3,0,1,0,0,3],[2,0,0,1,0,2,4,0,0,2,2,0,3,0,0,2],[0,2,3,0,4,0,0,4,2,0,0,4,0,4,2,0] >;
Q8.4S4 in GAP, Magma, Sage, TeX
Q_8._4S_4
% in TeX
G:=Group("Q8.4S4"); // GroupNames label
G:=SmallGroup(192,987);
// by ID
G=gap.SmallGroup(192,987);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,680,2102,1059,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=e^3=1,b^2=c^2=d^2=f^2=a^2,b*a*b^-1=f*a*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^-1*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a^2*d,f*e*f^-1=e^-1>;
// generators/relations
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