p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊7D4, C42.35C23, C4.1222+ 1+4, C2.52D42, C8⋊9D4⋊5C2, C8.31(C2×D4), D4⋊5D4⋊3C2, C4⋊D8⋊34C2, C8⋊3D4⋊18C2, C8⋊7D4⋊30C2, C8⋊D4⋊28C2, C4⋊C8⋊27C22, C4⋊C4.356D4, Q8⋊6D4⋊2C2, D4.20(C2×D4), Q8.18(C2×D4), D4⋊D4⋊34C2, C22⋊D8⋊27C2, (C2×D4).161D4, C2.38(D4○D8), C4⋊D4⋊9C22, (C2×C8).85C23, C22⋊C4.40D4, (C4×Q8)⋊19C22, C4.82(C22×D4), C8⋊C4⋊18C22, C2.D8⋊34C22, D4.7D4⋊34C2, C4⋊C4.207C23, C4⋊1D4⋊10C22, C22⋊C8⋊23C22, (C2×C4).466C24, Q8.D4⋊35C2, (C22×C8)⋊25C22, (C2×Q16)⋊46C22, (C2×D8).79C22, C23.100(C2×D4), SD16⋊C4⋊29C2, D4⋊C4⋊37C22, Q8⋊C4⋊39C22, (C2×SD16)⋊27C22, (C4×D4).142C22, (C2×D4).206C23, (C2×Q8).386C23, C22⋊Q8.56C22, C4.4D4.51C22, C22.726(C22×D4), C2.76(D8⋊C22), (C22×C4).1118C23, (C22×D4).398C22, (C2×M4(2)).101C22, (C2×C4○D8)⋊24C2, (C2×C8⋊C22)⋊29C2, (C2×C4).590(C2×D4), (C2×C4○D4)⋊14C22, SmallGroup(128,2000)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊7D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, cac-1=dad=a-1, cbc-1=a4b, bd=db, dcd=c-1 >
Subgroups: 584 in 255 conjugacy classes, 94 normal (84 characteristic)
C1, C2, C2, C4, C4, 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, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C8⋊9D4, SD16⋊C4, C22⋊D8, D4⋊D4, D4.7D4, C4⋊D8, Q8.D4, C8⋊7D4, C8⋊D4, C8⋊3D4, D4⋊5D4, Q8⋊6D4, C2×C4○D8, C2×C8⋊C22, SD16⋊7D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8⋊C22, D4○D8, SD16⋊7D4
Character table of SD16⋊7D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 2 | 0 | 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 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)
(1 24 12 31)(2 23 13 30)(3 22 14 29)(4 21 15 28)(5 20 16 27)(6 19 9 26)(7 18 10 25)(8 17 11 32)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)
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), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31), (1,24,12,31)(2,23,13,30)(3,22,14,29)(4,21,15,28)(5,20,16,27)(6,19,9,26)(7,18,10,25)(8,17,11,32), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31)>;
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), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31), (1,24,12,31)(2,23,13,30)(3,22,14,29)(4,21,15,28)(5,20,16,27)(6,19,9,26)(7,18,10,25)(8,17,11,32), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31) );
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)], [(1,16),(2,11),(3,14),(4,9),(5,12),(6,15),(7,10),(8,13),(17,26),(18,29),(19,32),(20,27),(21,30),(22,25),(23,28),(24,31)], [(1,24,12,31),(2,23,13,30),(3,22,14,29),(4,21,15,28),(5,20,16,27),(6,19,9,26),(7,18,10,25),(8,17,11,32)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31)]])
Matrix representation of SD16⋊7D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,3,3,0,0,0,0,14,3,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1] >;
SD16⋊7D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_7D_4 % in TeX
G:=Group("SD16:7D4"); // GroupNames label
G:=SmallGroup(128,2000);
// by ID
G=gap.SmallGroup(128,2000);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,346,2804,1411,375,172]);
// 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,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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