Aliases: Q16.A4, SL2(F3).11D4, 2+ 1+4.2C6, D4oD8:C3, C8oD4.C6, C8.A4:3C2, C8.2(C2xA4), C2.9(D4xA4), Q8.A4:4C2, Q8.3(C3xD4), Q8.4(C2xA4), C4.4(C22xA4), C4.A4.15C22, C4oD4.1(C2xC6), SmallGroup(192,1017)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16.A4
G = < a,b,c,d,e | a8=e3=1, b2=c2=d2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >
Subgroups: 291 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2xC4, D4, Q8, Q8, C23, C12, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C4oD4, C4oD4, C24, SL2(F3), C3xQ8, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, C3xQ16, C4.A4, C4.A4, D4oD8, C8.A4, Q8.A4, Q16.A4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2xC6, C3xD4, C2xA4, C22xA4, D4xA4, Q16.A4
Character table of Q16.A4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 12 | 12 | 4 | 4 | 2 | 4 | 4 | 6 | 4 | 4 | 2 | 2 | 12 | 8 | 8 | 16 | 16 | 16 | 16 | 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 | 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 | -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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | -2 | 0 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xD4 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | -2 | 0 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xD4 |
| ρ16 | 3 | 3 | -1 | 1 | -1 | 0 | 0 | 3 | -3 | 3 | -1 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ17 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ18 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | 3 | -3 | -3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ19 | 3 | 3 | -1 | -1 | 1 | 0 | 0 | 3 | 3 | -3 | -1 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ26 | 6 | 6 | 2 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4xA4 |
►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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 45 37 41)(34 44 38 48)(35 43 39 47)(36 42 40 46)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 47 37 43)(34 48 38 44)(35 41 39 45)(36 42 40 46)
(1 26 43)(2 27 44)(3 28 45)(4 29 46)(5 30 47)(6 31 48)(7 32 41)(8 25 42)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(13 17 37)(14 18 38)(15 19 39)(16 20 40)
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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,45,37,41)(34,44,38,48)(35,43,39,47)(36,42,40,46), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,47,37,43)(34,48,38,44)(35,41,39,45)(36,42,40,46), (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(13,17,37)(14,18,38)(15,19,39)(16,20,40)>;
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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,45,37,41)(34,44,38,48)(35,43,39,47)(36,42,40,46), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,23,21,19)(18,24,22,20)(25,27,29,31)(26,28,30,32)(33,47,37,43)(34,48,38,44)(35,41,39,45)(36,42,40,46), (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,41)(8,25,42)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(13,17,37)(14,18,38)(15,19,39)(16,20,40) );
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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(33,45,37,41),(34,44,38,48),(35,43,39,47),(36,42,40,46)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25),(33,45,37,41),(34,46,38,42),(35,47,39,43),(36,48,40,44)], [(1,11,5,15),(2,12,6,16),(3,13,7,9),(4,14,8,10),(17,23,21,19),(18,24,22,20),(25,27,29,31),(26,28,30,32),(33,47,37,43),(34,48,38,44),(35,41,39,45),(36,42,40,46)], [(1,26,43),(2,27,44),(3,28,45),(4,29,46),(5,30,47),(6,31,48),(7,32,41),(8,25,42),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(13,17,37),(14,18,38),(15,19,39),(16,20,40)]])
Matrix representation of Q16.A4 ►in GL4(F7) generated by
| 4 | 5 | 3 | 6 |
| 4 | 2 | 4 | 4 |
| 3 | 3 | 0 | 1 |
| 2 | 5 | 6 | 2 |
| 6 | 5 | 3 | 0 |
| 0 | 0 | 6 | 2 |
| 4 | 4 | 5 | 6 |
| 2 | 5 | 6 | 3 |
| 6 | 2 | 0 | 5 |
| 5 | 0 | 4 | 3 |
| 2 | 5 | 4 | 3 |
| 6 | 6 | 4 | 4 |
| 3 | 6 | 6 | 3 |
| 4 | 0 | 3 | 2 |
| 1 | 6 | 0 | 5 |
| 3 | 3 | 2 | 4 |
| 6 | 2 | 6 | 3 |
| 6 | 3 | 3 | 2 |
| 0 | 0 | 2 | 0 |
| 5 | 5 | 1 | 1 |
G:=sub<GL(4,GF(7))| [4,4,3,2,5,2,3,5,3,4,0,6,6,4,1,2],[6,0,4,2,5,0,4,5,3,6,5,6,0,2,6,3],[6,5,2,6,2,0,5,6,0,4,4,4,5,3,3,4],[3,4,1,3,6,0,6,3,6,3,0,2,3,2,5,4],[6,6,0,5,2,3,0,5,6,3,2,1,3,2,0,1] >;
Q16.A4 in GAP, Magma, Sage, TeX
Q_{16}.A_4 % in TeX
G:=Group("Q16.A4"); // GroupNames label
G:=SmallGroup(192,1017);
// by ID
G=gap.SmallGroup(192,1017);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,197,680,3027,1522,248,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=e^3=1,b^2=c^2=d^2=a^4,b*a*b^-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,d*c*d^-1=a^4*c,e*c*e^-1=a^4*c*d,e*d*e^-1=c>;
// generators/relations
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