p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊7D4, C42.35C23, C4.1222+ (1+4), C8⋊9D4⋊5C2, C2.52(D42), C8.31(C2×D4), D4⋊5D4⋊3C2, C8⋊7D4⋊30C2, C8⋊D4⋊28C2, C4⋊D8⋊34C2, C8⋊3D4⋊18C2, C4⋊C8⋊27C22, C4⋊C4.356D4, Q8⋊6D4⋊2C2, D4.20(C2×D4), Q8.18(C2×D4), C22⋊D8⋊27C2, D4⋊D4⋊34C2, (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⋊1D4⋊10C22, C4⋊C4.207C23, 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, (C2×D4).206C23, (C4×D4).142C22, (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
Subgroups: 584 in 255 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×23], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×2], D4 [×22], Q8 [×2], Q8 [×3], C23 [×2], C23 [×9], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×7], SD16 [×4], SD16 [×5], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×5], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×9], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4 [×3], C4⋊D4 [×3], C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4○D8 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×3], C8⋊9D4, SD16⋊C4, C22⋊D8, D4⋊D4 [×2], 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 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ (1+4), D42, D8⋊C22, D4○D8, SD16⋊7D4
Generators and relations
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 >
►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 27)(2 30)(3 25)(4 28)(5 31)(6 26)(7 29)(8 32)(9 20)(10 23)(11 18)(12 21)(13 24)(14 19)(15 22)(16 17)
(1 24 31 13)(2 23 32 12)(3 22 25 11)(4 21 26 10)(5 20 27 9)(6 19 28 16)(7 18 29 15)(8 17 30 14)
(2 8)(3 7)(4 6)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 29)(26 28)(30 32)
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), (1,27)(2,30)(3,25)(4,28)(5,31)(6,26)(7,29)(8,32)(9,20)(10,23)(11,18)(12,21)(13,24)(14,19)(15,22)(16,17), (1,24,31,13)(2,23,32,12)(3,22,25,11)(4,21,26,10)(5,20,27,9)(6,19,28,16)(7,18,29,15)(8,17,30,14), (2,8)(3,7)(4,6)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,29)(26,28)(30,32)>;
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), (1,27)(2,30)(3,25)(4,28)(5,31)(6,26)(7,29)(8,32)(9,20)(10,23)(11,18)(12,21)(13,24)(14,19)(15,22)(16,17), (1,24,31,13)(2,23,32,12)(3,22,25,11)(4,21,26,10)(5,20,27,9)(6,19,28,16)(7,18,29,15)(8,17,30,14), (2,8)(3,7)(4,6)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,29)(26,28)(30,32) );
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)], [(1,27),(2,30),(3,25),(4,28),(5,31),(6,26),(7,29),(8,32),(9,20),(10,23),(11,18),(12,21),(13,24),(14,19),(15,22),(16,17)], [(1,24,31,13),(2,23,32,12),(3,22,25,11),(4,21,26,10),(5,20,27,9),(6,19,28,16),(7,18,29,15),(8,17,30,14)], [(2,8),(3,7),(4,6),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,29),(26,28),(30,32)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
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