Aliases: Q16.A4, SL2(𝔽3).11D4, 2+ 1+4.2C6, D4○D8⋊C3, C8○D4.C6, C8.A4⋊3C2, C8.2(C2×A4), C2.9(D4×A4), Q8.A4⋊4C2, Q8.3(C3×D4), Q8.4(C2×A4), C4.4(C22×A4), C4.A4.15C22, C4○D4.1(C2×C6), 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, C2×C4, D4, Q8, Q8, C23, C12, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C4○D4, C4○D4, C24, SL2(𝔽3), C3×Q8, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C3×Q16, C4.A4, C4.A4, D4○D8, C8.A4, Q8.A4, Q16.A4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, C3×D4, C2×A4, C22×A4, D4×A4, 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 C3×D4 |
| 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 C3×D4 |
| 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 C2×A4 |
| 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 C2×A4 |
| 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 C2×A4 |
| 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 D4×A4 |
►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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