p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊5D4, C42.32C23, C4.1192+ 1+4, D42⋊4C2, C8⋊9(C2×D4), C2.49D42, Q8⋊3(C2×D4), C8⋊9D4⋊2C2, C8⋊D4⋊26C2, C8⋊3D4⋊17C2, C8⋊7D4⋊29C2, C4⋊C8⋊24C22, C4⋊C4.147D4, Q8⋊5D4⋊1C2, D4.17(C2×D4), C22⋊D8⋊26C2, D4⋊D4⋊32C2, C4⋊SD16⋊17C2, Q8⋊D4⋊14C2, (C2×D4).306D4, (C2×D8)⋊28C22, C4⋊D4⋊8C22, (C2×C8).82C23, (C4×Q8)⋊17C22, C4.79(C22×D4), C8⋊C4⋊15C22, C2.D8⋊32C22, C22⋊Q8⋊8C22, D4.2D4⋊35C2, C22⋊SD16⋊15C2, C4⋊C4.204C23, C22⋊C8⋊20C22, C22⋊3(C8⋊C22), (C2×C4).463C24, (C22×C8)⋊23C22, C22⋊C4.157D4, (C22×SD16)⋊6C2, C23.460(C2×D4), SD16⋊C4⋊27C2, D4⋊C4⋊35C22, C2.56(D4○SD16), Q8⋊C4⋊36C22, (C2×SD16)⋊26C22, (C4×D4).140C22, (C2×D4).203C23, C4.4D4⋊12C22, C4⋊1D4.74C22, (C2×Q8).385C23, (C22×Q8)⋊22C22, (C2×M4(2))⋊20C22, (C22×C4).317C23, C22.723(C22×D4), (C22×D4).395C22, (C2×C8⋊C22)⋊27C2, (C2×C4).587(C2×D4), C2.71(C2×C8⋊C22), (C2×C4○D4)⋊12C22, SmallGroup(128,1997)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 648 in 273 conjugacy classes, 96 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C8⋊9D4, SD16⋊C4, C22⋊D8, Q8⋊D4, D4⋊D4, C22⋊SD16, C4⋊SD16, D4.2D4, C8⋊7D4, C8⋊D4, C8⋊3D4, D42, Q8⋊5D4, C22×SD16, C2×C8⋊C22, SD16⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, D42, C2×C8⋊C22, D4○SD16, SD16⋊D4
Character table of SD16⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 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 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ8 | 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 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 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 | 1 | -1 | 1 | linear of order 2 |
| ρ10 | 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 | 1 | 1 | -1 | linear of order 2 |
| ρ11 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ12 | 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 | 1 | -1 | -1 | linear of order 2 |
| ρ13 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ14 | 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 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 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 | -1 | 1 | 1 | linear of order 2 |
| ρ16 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ27 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | 0 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
►On 32 points
(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)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 26 23 12)(2 25 24 11)(3 32 17 10)(4 31 18 9)(5 30 19 16)(6 29 20 15)(7 28 21 14)(8 27 22 13)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,26,23,12)(2,25,24,11)(3,32,17,10)(4,31,18,9)(5,30,19,16)(6,29,20,15)(7,28,21,14)(8,27,22,13), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;
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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,26,23,12)(2,25,24,11)(3,32,17,10)(4,31,18,9)(5,30,19,16)(6,29,20,15)(7,28,21,14)(8,27,22,13), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );
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)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,26,23,12),(2,25,24,11),(3,32,17,10),(4,31,18,9),(5,30,19,16),(6,29,20,15),(7,28,21,14),(8,27,22,13)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)]])
Matrix representation of SD16⋊D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 16 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 6 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 5 | 7 | 7 |
| 0 | 0 | 12 | 1 | 0 | 10 |
| 0 | 0 | 6 | 6 | 6 | 12 |
| 0 | 0 | 0 | 11 | 12 | 11 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 7 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 5 | 7 | 7 |
| 0 | 0 | 12 | 1 | 0 | 10 |
| 0 | 0 | 6 | 6 | 6 | 12 |
| 0 | 0 | 0 | 11 | 12 | 11 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,16,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[2,6,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11],[2,7,0,0,0,0,2,15,0,0,0,0,0,0,16,12,6,0,0,0,5,1,6,11,0,0,7,0,6,12,0,0,7,10,12,11] >;
SD16⋊D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes D_4 % in TeX
G:=Group("SD16:D4"); // GroupNames label
G:=SmallGroup(128,1997);
// by ID
G=gap.SmallGroup(128,1997);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,c*a*c^-1=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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