p-group, metabelian, nilpotent (class 4), monomial
Aliases: Q8○D16, D4○Q32, D4.14D8, Q8.14D8, C8.18C24, C16.5C23, D8.7C23, SD32.C22, D16.4C22, Q32.4C22, Q16.7C23, M4(2).23D4, M5(2).14C22, Q8○D8⋊6C2, C4○D16⋊6C2, D4○C16⋊5C2, C8.17(C2×D4), C4.51(C2×D8), (C2×Q32)⋊13C2, C4○D4.37D4, Q32⋊C2⋊6C2, C22.8(C2×D8), C2.33(C22×D8), C4.24(C22×D4), (C2×C8).296C23, (C2×C16).35C22, C8○D4.14C22, C4○D8.11C22, (C2×Q16).97C22, (C2×C4).186(C2×D4), SmallGroup(128,2149)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8○D16
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c7 >
Subgroups: 352 in 175 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C16, C16, C2×C8, M4(2), D8, SD16, Q16, Q16, C2×Q8, C4○D4, C4○D4, C2×C16, M5(2), D16, SD32, Q32, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, D4○C16, C2×Q32, C4○D16, Q32⋊C2, Q8○D8, Q8○D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C22×D8, Q8○D16
►On 64 points
(1 26 9 18)(2 27 10 19)(3 28 11 20)(4 29 12 21)(5 30 13 22)(6 31 14 23)(7 32 15 24)(8 17 16 25)(33 63 41 55)(34 64 42 56)(35 49 43 57)(36 50 44 58)(37 51 45 59)(38 52 46 60)(39 53 47 61)(40 54 48 62)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 33)(32 34)
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 27 9 19)(2 26 10 18)(3 25 11 17)(4 24 12 32)(5 23 13 31)(6 22 14 30)(7 21 15 29)(8 20 16 28)(33 62 41 54)(34 61 42 53)(35 60 43 52)(36 59 44 51)(37 58 45 50)(38 57 46 49)(39 56 47 64)(40 55 48 63)
G:=sub<Sym(64)| (1,26,9,18)(2,27,10,19)(3,28,11,20)(4,29,12,21)(5,30,13,22)(6,31,14,23)(7,32,15,24)(8,17,16,25)(33,63,41,55)(34,64,42,56)(35,49,43,57)(36,50,44,58)(37,51,45,59)(38,52,46,60)(39,53,47,61)(40,54,48,62), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,27,9,19)(2,26,10,18)(3,25,11,17)(4,24,12,32)(5,23,13,31)(6,22,14,30)(7,21,15,29)(8,20,16,28)(33,62,41,54)(34,61,42,53)(35,60,43,52)(36,59,44,51)(37,58,45,50)(38,57,46,49)(39,56,47,64)(40,55,48,63)>;
G:=Group( (1,26,9,18)(2,27,10,19)(3,28,11,20)(4,29,12,21)(5,30,13,22)(6,31,14,23)(7,32,15,24)(8,17,16,25)(33,63,41,55)(34,64,42,56)(35,49,43,57)(36,50,44,58)(37,51,45,59)(38,52,46,60)(39,53,47,61)(40,54,48,62), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,33)(32,34), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,27,9,19)(2,26,10,18)(3,25,11,17)(4,24,12,32)(5,23,13,31)(6,22,14,30)(7,21,15,29)(8,20,16,28)(33,62,41,54)(34,61,42,53)(35,60,43,52)(36,59,44,51)(37,58,45,50)(38,57,46,49)(39,56,47,64)(40,55,48,63) );
G=PermutationGroup([[(1,26,9,18),(2,27,10,19),(3,28,11,20),(4,29,12,21),(5,30,13,22),(6,31,14,23),(7,32,15,24),(8,17,16,25),(33,63,41,55),(34,64,42,56),(35,49,43,57),(36,50,44,58),(37,51,45,59),(38,52,46,60),(39,53,47,61),(40,54,48,62)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,33),(32,34)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,27,9,19),(2,26,10,18),(3,25,11,17),(4,24,12,32),(5,23,13,31),(6,22,14,30),(7,21,15,29),(8,20,16,28),(33,62,41,54),(34,61,42,53),(35,60,43,52),(36,59,44,51),(37,58,45,50),(38,57,46,49),(39,56,47,64),(40,55,48,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 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 | Q8○D16 |
| kernel | Q8○D16 | D4○C16 | C2×Q32 | C4○D16 | Q32⋊C2 | Q8○D8 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 3 | 1 | 6 | 2 | 4 |
Matrix representation of Q8○D16 ►in GL4(𝔽17) generated by
| 7 | 0 | 16 | 2 |
| 0 | 7 | 16 | 1 |
| 1 | 15 | 10 | 0 |
| 1 | 16 | 0 | 10 |
| 1 | 15 | 10 | 0 |
| 1 | 16 | 0 | 10 |
| 7 | 0 | 16 | 2 |
| 0 | 7 | 16 | 1 |
| 15 | 9 | 0 | 0 |
| 4 | 7 | 0 | 0 |
| 0 | 0 | 15 | 9 |
| 0 | 0 | 4 | 7 |
| 15 | 5 | 2 | 5 |
| 9 | 2 | 13 | 15 |
| 2 | 5 | 15 | 5 |
| 13 | 15 | 9 | 2 |
G:=sub<GL(4,GF(17))| [7,0,1,1,0,7,15,16,16,16,10,0,2,1,0,10],[1,1,7,0,15,16,0,7,10,0,16,16,0,10,2,1],[15,4,0,0,9,7,0,0,0,0,15,4,0,0,9,7],[15,9,2,13,5,2,5,15,2,13,15,9,5,15,5,2] >;
Q8○D16 in GAP, Magma, Sage, TeX
Q_8\circ D_{16} % in TeX
G:=Group("Q8oD16"); // GroupNames label
G:=SmallGroup(128,2149);
// by ID
G=gap.SmallGroup(128,2149);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,456,521,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=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^-1=a^2*c^7>;
// generators/relations