Aliases: D8.A4, 2- 1+4⋊C6, SL2(𝔽3).13D4, Q8○D8⋊C3, C8.A4⋊4C2, C8○D4⋊2C6, C8.3(C2×A4), D4.A4⋊5C2, D4.3(C2×A4), C2.11(D4×A4), Q8.5(C3×D4), C4.6(C22×A4), C4.A4.17C22, C4○D4.3(C2×C6), SmallGroup(192,1019)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.A4
G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >
Subgroups: 259 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C12, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C4○D4, C4○D4, C24, SL2(𝔽3), C3×D4, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C3×D8, C2×SL2(𝔽3), C4.A4, Q8○D8, C8.A4, D4.A4, D8.A4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, C3×D4, C2×A4, C22×A4, D4×A4, D8.A4
Character table of D8.A4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | 2 | 6 | 12 | 12 | 4 | 4 | 16 | 16 | 16 | 16 | 2 | 2 | 12 | 8 | 8 | 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 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ3 | ζ32 | -1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ32 | ζ3 | -1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | -1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | -1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 0 | 0 | -2 | -1-√-3 | -1+√-3 | -2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | 2 | 0 | 0 | -2 | -1+√-3 | -1-√-3 | -2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 3 | 3 | -3 | -3 | -1 | 0 | 0 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ17 | 3 | 3 | 3 | 3 | -1 | 0 | 0 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ18 | 3 | 3 | -3 | 3 | -1 | 0 | 0 | 3 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ19 | 3 | 3 | 3 | -3 | -1 | 0 | 0 | 3 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 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 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ26 | 6 | 6 | 0 | 0 | 2 | 0 | 0 | -6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×A4 |
►On 32 points
Generators in S32
(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 29 21 25)(18 30 22 26)(19 31 23 27)(20 32 24 28)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)
(9 30 22)(10 31 23)(11 32 24)(12 25 17)(13 26 18)(14 27 19)(15 28 20)(16 29 21)
G:=sub<Sym(32)| (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,29,21,25)(18,30,22,26)(19,31,23,27)(20,32,24,28), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,30,22)(10,31,23)(11,32,24)(12,25,17)(13,26,18)(14,27,19)(15,28,20)(16,29,21)>;
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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31), (1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(17,29,21,25)(18,30,22,26)(19,31,23,27)(20,32,24,28), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,30,22)(10,31,23)(11,32,24)(12,25,17)(13,26,18)(14,27,19)(15,28,20)(16,29,21) );
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)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(17,19),(20,24),(21,23),(25,27),(28,32),(29,31)], [(1,9,5,13),(2,10,6,14),(3,11,7,15),(4,12,8,16),(17,29,21,25),(18,30,22,26),(19,31,23,27),(20,32,24,28)], [(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17)], [(9,30,22),(10,31,23),(11,32,24),(12,25,17),(13,26,18),(14,27,19),(15,28,20),(16,29,21)]])
Matrix representation of D8.A4 ►in GL4(𝔽7) generated by
| 1 | 1 | 3 | 6 |
| 2 | 4 | 0 | 5 |
| 5 | 4 | 3 | 4 |
| 2 | 0 | 3 | 5 |
| 6 | 0 | 0 | 0 |
| 6 | 6 | 6 | 3 |
| 5 | 6 | 4 | 5 |
| 1 | 2 | 1 | 5 |
| 6 | 2 | 1 | 4 |
| 5 | 4 | 4 | 4 |
| 6 | 0 | 0 | 1 |
| 6 | 2 | 0 | 4 |
| 6 | 6 | 3 | 5 |
| 2 | 1 | 6 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 3 | 2 |
| 5 | 5 | 6 | 3 |
| 2 | 1 | 5 | 0 |
| 0 | 3 | 4 | 3 |
| 5 | 1 | 2 | 2 |
G:=sub<GL(4,GF(7))| [1,2,5,2,1,4,4,0,3,0,3,3,6,5,4,5],[6,6,5,1,0,6,6,2,0,6,4,1,0,3,5,5],[6,5,6,6,2,4,0,2,1,4,0,0,4,4,1,4],[6,2,0,0,6,1,0,0,3,6,5,3,5,0,3,2],[5,2,0,5,5,1,3,1,6,5,4,2,3,0,3,2] >;
D8.A4 in GAP, Magma, Sage, TeX
D_8.A_4
% in TeX
G:=Group("D8.A4"); // GroupNames label
G:=SmallGroup(192,1019);
// by ID
G=gap.SmallGroup(192,1019);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,3027,1522,248,438,172,775,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=e^3=1,c^2=d^2=a^4,b*a*b=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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