direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4×SD16, C42.448C23, C4.1352+ 1+4, D42.3C2, C2.65D42, C8⋊14(C2×D4), (D4×Q8)⋊6C2, Q8⋊5(C2×D4), (C8×D4)⋊30C2, C4⋊2(C2×SD16), C8⋊8D4⋊33C2, C8⋊5D4⋊26C2, C4⋊C8⋊77C22, (C4×C8)⋊44C22, C4⋊C4.254D4, D4.27(C2×D4), C4⋊Q8⋊17C22, C4⋊SD16⋊40C2, Q8⋊D4⋊32C2, (C4×SD16)⋊35C2, (C2×D4).348D4, C22⋊C4.93D4, (C4×Q8)⋊22C22, C22⋊2(C2×SD16), C4.95(C22×D4), C4.Q8⋊69C22, C22⋊SD16⋊33C2, D4.D4⋊41C2, C4⋊C4.220C23, C22⋊C8⋊69C22, (C2×C8).344C23, (C2×C4).479C24, (C22×C8)⋊38C22, C23.465(C2×D4), C22⋊Q8⋊13C22, D4⋊C4⋊46C22, C2.63(D4○SD16), Q8⋊C4⋊85C22, (C22×SD16)⋊28C2, (C2×SD16)⋊49C22, (C2×D4).418C23, (C4×D4).323C22, C4⋊D4.65C22, C4⋊1D4.78C22, (C2×Q8).200C23, (C22×Q8)⋊25C22, C2.27(C22×SD16), C22.739(C22×D4), (C22×C4).1123C23, (C22×D4).403C22, (C2×D4)○(C2×SD16), (C2×C4).161(C2×D4), SmallGroup(128,2013)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4×SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 624 in 273 conjugacy classes, 104 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, 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, C2×C8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C2×SD16, C22×D4, C22×D4, C22×Q8, C8×D4, C4×SD16, Q8⋊D4, C22⋊SD16, C4⋊SD16, D4.D4, C8⋊8D4, C8⋊5D4, D42, D4×Q8, C22×SD16, D4×SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C2×SD16, C22×D4, 2+ 1+4, D42, C22×SD16, D4○SD16, D4×SD16
►On 32 points
(1 14 20 32)(2 15 21 25)(3 16 22 26)(4 9 23 27)(5 10 24 28)(6 11 17 29)(7 12 18 30)(8 13 19 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
G:=sub<Sym(32)| (1,14,20,32)(2,15,21,25)(3,16,22,26)(4,9,23,27)(5,10,24,28)(6,11,17,29)(7,12,18,30)(8,13,19,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30)>;
G:=Group( (1,14,20,32)(2,15,21,25)(3,16,22,26)(4,9,23,27)(5,10,24,28)(6,11,17,29)(7,12,18,30)(8,13,19,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30) );
G=PermutationGroup([[(1,14,20,32),(2,15,21,25),(3,16,22,26),(4,9,23,27),(5,10,24,28),(6,11,17,29),(7,12,18,30),(8,13,19,31)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)], [(1,20),(2,23),(3,18),(4,21),(5,24),(6,19),(7,22),(8,17),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | SD16 | 2+ 1+4 | D4○SD16 |
| kernel | D4×SD16 | C8×D4 | C4×SD16 | Q8⋊D4 | C22⋊SD16 | C4⋊SD16 | D4.D4 | C8⋊8D4 | C8⋊5D4 | D42 | D4×Q8 | C22×SD16 | C22⋊C4 | C4⋊C4 | SD16 | C2×D4 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of D4×SD16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 1 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 16 |
| 10 | 10 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,15,16],[10,12,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[1,16,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;
D4×SD16 in GAP, Magma, Sage, TeX
D_4\times {\rm SD}_{16} % in TeX
G:=Group("D4xSD16"); // GroupNames label
G:=SmallGroup(128,2013);
// by ID
G=gap.SmallGroup(128,2013);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations