p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8○Q16, D8.14D4, Q16.14D4, SD16.1D4, C42.458C23, M4(2).18C23, C22.62+ (1+4), 2- (1+4).5C22, Q8○D8⋊4C2, C8○D8⋊11C2, C2.77(D42), C4≀C2.C22, C8.14(C2×D4), D4.36(C2×D4), Q8.36(C2×D4), C4⋊Q16⋊23C2, D4.5D4⋊7C2, (C2×C4).26C24, C8○D4.6C22, D4.10D4⋊7C2, (C2×Q8).9C23, C8.C22.C22, (C2×C8).290C23, (C4×C8).192C22, C4○D4.15C23, C4○D8.30C22, C4.107(C22×D4), C4⋊Q8.140C22, (C2×Q16).82C22, C8.C4.23C22, C4.10D4.4C22, SmallGroup(128,2025)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 412 in 222 conjugacy classes, 92 normal (8 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×10], C22, C22 [×4], C8 [×4], C8 [×4], C2×C4, C2×C4 [×17], D4 [×4], D4 [×8], Q8 [×4], Q8 [×12], C42, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], SD16 [×4], SD16 [×8], Q16 [×2], Q16 [×16], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×4], C4○D4 [×12], C4×C8, C4.10D4 [×4], C4≀C2 [×4], C8.C4 [×2], C4⋊Q8 [×2], C8○D4 [×4], C2×Q16 [×4], C2×Q16 [×4], C4○D8 [×2], C4○D8 [×4], C8.C22 [×8], C8.C22 [×8], 2- (1+4) [×4], C8○D8 [×2], D4.10D4 [×4], D4.5D4 [×4], C4⋊Q16, Q8○D8 [×4], D8○Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ (1+4), D42, D8○Q16
Generators and relations
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a4c3 >
►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 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)
(1 2 3 4 5 6 7 8)(9 16 15 14 13 12 11 10)(17 24 23 22 21 20 19 18)(25 26 27 28 29 30 31 32)
(1 20 5 24)(2 21 6 17)(3 22 7 18)(4 23 8 19)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,24,23,22,21,20,19,18)(25,26,27,28,29,30,31,32), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)>;
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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,16)(7,15)(8,14)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32), (1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,24,23,22,21,20,19,18)(25,26,27,28,29,30,31,32), (1,20,5,24)(2,21,6,17)(3,22,7,18)(4,23,8,19)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31) );
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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,16),(7,15),(8,14),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32)], [(1,2,3,4,5,6,7,8),(9,16,15,14,13,12,11,10),(17,24,23,22,21,20,19,18),(25,26,27,28,29,30,31,32)], [(1,20,5,24),(2,21,6,17),(3,22,7,18),(4,23,8,19),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31)])
Matrix representation ►G ⊆ GL4(𝔽7) generated by
| 5 | 0 | 2 | 6 |
| 6 | 5 | 5 | 6 |
| 6 | 1 | 1 | 2 |
| 2 | 2 | 6 | 2 |
| 4 | 2 | 5 | 3 |
| 3 | 3 | 3 | 6 |
| 6 | 4 | 6 | 3 |
| 4 | 5 | 0 | 1 |
| 5 | 2 | 4 | 5 |
| 1 | 5 | 4 | 2 |
| 1 | 1 | 0 | 5 |
| 5 | 2 | 1 | 3 |
| 4 | 2 | 2 | 1 |
| 1 | 4 | 1 | 3 |
| 0 | 4 | 5 | 5 |
| 2 | 4 | 1 | 1 |
G:=sub<GL(4,GF(7))| [5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[4,3,6,4,2,3,4,5,5,3,6,0,3,6,3,1],[5,1,1,5,2,5,1,2,4,4,0,1,5,2,5,3],[4,1,0,2,2,4,4,4,2,1,5,1,1,3,5,1] >;
Character table of D8○Q16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 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 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ (1+4) |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
In GAP, Magma, Sage, TeX
D_8\circ Q_{16} % in TeX
G:=Group("D8oQ16"); // GroupNames label
G:=SmallGroup(128,2025);
// by ID
G=gap.SmallGroup(128,2025);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,352,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^3>;
// generators/relations