p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊1D4, C42.441C23, C4.1282+ 1+4, D42⋊6C2, C8⋊3(C2×D4), C2.58D42, Q8⋊4(C2×D4), C8⋊6D4⋊5C2, C8⋊4D4⋊21C2, C8⋊2D4⋊21C2, C4⋊D8⋊36C2, C4⋊C8⋊30C22, C4⋊3(C8⋊C22), C4⋊C4.360D4, (C4×C8)⋊31C22, Q8⋊6D4⋊4C2, D4.23(C2×D4), (C2×D8)⋊7C22, D4⋊D4⋊38C2, C22⋊D8⋊29C2, C4⋊SD16⋊19C2, (C4×SD16)⋊19C2, (C2×D4).310D4, C2.40(D4○D8), C22⋊C4.43D4, (C4×Q8)⋊20C22, C4.88(C22×D4), C4.Q8⋊52C22, C4⋊1D4⋊11C22, C4⋊C4.213C23, C4⋊D4⋊10C22, C22⋊C8⋊26C22, (C2×C8).284C23, (C2×C4).472C24, C23.314(C2×D4), D4⋊C4⋊38C22, Q8⋊C4⋊71C22, (C2×SD16)⋊29C22, (C4×D4).146C22, (C2×D4).211C23, (C2×Q8).389C23, (C2×M4(2))⋊24C22, (C22×C4).322C23, C22.732(C22×D4), (C22×D4).400C22, (C2×C8⋊C22)⋊32C2, (C2×C4).156(C2×D4), C2.73(C2×C8⋊C22), (C2×C4○D4)⋊16C22, SmallGroup(128,2006)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16⋊1D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a3, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 688 in 278 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, 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×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C4⋊1D4, C4⋊1D4, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C22×D4, C2×C4○D4, C8⋊6D4, C4×SD16, C22⋊D8, D4⋊D4, C4⋊D8, C4⋊SD16, C8⋊2D4, C8⋊4D4, D42, Q8⋊6D4, C2×C8⋊C22, SD16⋊1D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, D42, C2×C8⋊C22, D4○D8, SD16⋊1D4
Character table of SD16⋊1D4
| 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 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| 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 | -2√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| 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 | 2√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
►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 29 24 15)(3 32 17 10)(4 27 18 13)(5 30 19 16)(6 25 20 11)(7 28 21 14)(8 31 22 9)
(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)(2,29,24,15)(3,32,17,10)(4,27,18,13)(5,30,19,16)(6,25,20,11)(7,28,21,14)(8,31,22,9), (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)(2,29,24,15)(3,32,17,10)(4,27,18,13)(5,30,19,16)(6,25,20,11)(7,28,21,14)(8,31,22,9), (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),(2,29,24,15),(3,32,17,10),(4,27,18,13),(5,30,19,16),(6,25,20,11),(7,28,21,14),(8,31,22,9)], [(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⋊1D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,3,3,0,0,0,0,14,3,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,3,3,0,0,0,0,14,3,0,0] >;
SD16⋊1D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_1D_4 % in TeX
G:=Group("SD16:1D4"); // GroupNames label
G:=SmallGroup(128,2006);
// by ID
G=gap.SmallGroup(128,2006);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^3,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export