p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊1D4, C42.441C23, C4.1282+ (1+4), C8⋊3(C2×D4), (D42)⋊6C2, Q8⋊4(C2×D4), C8⋊6D4⋊5C2, C2.58(D42), C8⋊2D4⋊21C2, C8⋊4D4⋊21C2, C4⋊D8⋊36C2, C4⋊C8⋊30C22, C4⋊3(C8⋊C22), (C4×C8)⋊31C22, C4⋊C4.360D4, 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
Subgroups: 688 in 278 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×30], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×13], D4 [×2], D4 [×31], Q8 [×2], Q8, C23 [×2], C23 [×15], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×10], SD16 [×4], SD16 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×22], C2×Q8, C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×4], Q8⋊C4, C4⋊C8, C4.Q8, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4 [×4], C4⋊D4 [×3], C4⋊1D4 [×2], C4⋊1D4, C2×M4(2) [×2], C2×D8 [×6], C2×SD16, C2×SD16 [×2], C8⋊C22 [×8], C22×D4 [×2], C22×D4, C2×C4○D4 [×2], C8⋊6D4, C4×SD16, C22⋊D8 [×2], D4⋊D4 [×2], C4⋊D8, C4⋊SD16, C8⋊2D4 [×2], C8⋊4D4, D42, Q8⋊6D4, C2×C8⋊C22 [×2], SD16⋊1D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8⋊C22 [×2], C22×D4 [×2], 2+ (1+4), D42, C2×C8⋊C22, D4○D8, SD16⋊1D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a3, dad=a-1, bc=cb, 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 21)(2 24)(3 19)(4 22)(5 17)(6 20)(7 23)(8 18)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 26 21 12)(2 29 22 15)(3 32 23 10)(4 27 24 13)(5 30 17 16)(6 25 18 11)(7 28 19 14)(8 31 20 9)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 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), (1,21)(2,24)(3,19)(4,22)(5,17)(6,20)(7,23)(8,18)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,26,21,12)(2,29,22,15)(3,32,23,10)(4,27,24,13)(5,30,17,16)(6,25,18,11)(7,28,19,14)(8,31,20,9), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(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), (1,21)(2,24)(3,19)(4,22)(5,17)(6,20)(7,23)(8,18)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,26,21,12)(2,29,22,15)(3,32,23,10)(4,27,24,13)(5,30,17,16)(6,25,18,11)(7,28,19,14)(8,31,20,9), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(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)], [(1,21),(2,24),(3,19),(4,22),(5,17),(6,20),(7,23),(8,18),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,26,21,12),(2,29,22,15),(3,32,23,10),(4,27,24,13),(5,30,17,16),(6,25,18,11),(7,28,19,14),(8,31,20,9)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
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