p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4○D16, Q8○Q32, D4.12D8, Q8.12D8, D16⋊5C22, C8.16C24, C16.3C23, Q32⋊7C22, D8.5C23, SD32⋊4C22, Q16.5C23, M4(2).21D4, M5(2)⋊8C22, D4○D8⋊5C2, D4○C16⋊3C2, C4○D16⋊4C2, C8.15(C2×D4), C4.49(C2×D8), (C2×D16)⋊13C2, C16⋊C22⋊5C2, (C2×C16)⋊5C22, C4○D4.35D4, C4○D8⋊1C22, C22.6(C2×D8), (C2×D8)⋊33C22, C2.31(C22×D8), C4.22(C22×D4), (C2×C8).294C23, C8○D4.12C22, (C2×C4).184(C2×D4), SmallGroup(128,2147)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4○D16
G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c7 >
Subgroups: 480 in 185 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C16, C16, C2×C8, M4(2), D8, D8, SD16, Q16, C2×D4, C4○D4, C4○D4, C2×C16, M5(2), D16, SD32, Q32, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, D4○C16, C2×D16, C4○D16, C16⋊C22, D4○D8, D4○D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C22×D8, D4○D16
►On 32 points
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)
(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 18)(2 17)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,18)(2,17)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)>;
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,18)(2,17)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19) );
G=PermutationGroup([[(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,30),(2,31),(3,32),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29)], [(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)], [(1,18),(2,17),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | 16E | ··· | 16J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 | 16 | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D4○D16 |
| kernel | D4○D16 | D4○C16 | C2×D16 | C4○D16 | C16⋊C22 | D4○D8 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 3 | 1 | 6 | 2 | 4 |
Matrix representation of D4○D16 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 16 | 0 | 16 | 0 |
| 0 | 16 | 0 | 15 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 1 |
| 16 | 0 | 16 | 0 |
| 4 | 11 | 0 | 0 |
| 6 | 4 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 6 | 4 |
| 3 | 14 | 6 | 11 |
| 14 | 14 | 11 | 11 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 3 | 3 |
G:=sub<GL(4,GF(17))| [0,1,0,16,16,0,1,0,0,0,0,16,0,0,1,0],[0,1,0,16,16,0,1,0,0,2,0,16,15,0,1,0],[4,6,0,0,11,4,0,0,0,0,4,6,0,0,11,4],[3,14,0,0,14,14,0,0,6,11,14,3,11,11,3,3] >;
D4○D16 in GAP, Magma, Sage, TeX
D_4\circ D_{16} % in TeX
G:=Group("D4oD16"); // GroupNames label
G:=SmallGroup(128,2147);
// by ID
G=gap.SmallGroup(128,2147);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,521,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^8=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^7>;
// generators/relations