p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4○Q32, Q8○D16, D4.14D8, Q8.14D8, C16.5C23, C8.18C24, D8.7C23, SD32.C22, D16.4C22, Q32.4C22, Q16.7C23, M4(2).23D4, M5(2).14C22, Q8○D8⋊6C2, D4○C16⋊5C2, C4○D16⋊6C2, C4.51(C2×D8), C8.17(C2×D4), (C2×Q32)⋊13C2, C4○D4.37D4, Q32⋊C2⋊6C2, C22.8(C2×D8), C4.24(C22×D4), C2.33(C22×D8), (C2×C16).35C22, (C2×C8).296C23, C4○D8.11C22, C8○D4.14C22, (C2×Q16).97C22, (C2×C4).186(C2×D4), SmallGroup(128,2149)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4○Q32
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 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8 [×2], SD16 [×6], Q16 [×6], Q16 [×6], C2×Q8 [×8], C4○D4, C4○D4 [×12], C2×C16 [×3], M5(2) [×3], D16, SD32 [×6], Q32 [×9], C8○D4, C2×Q16 [×6], C4○D8 [×6], C8.C22 [×6], 2- 1+4 [×2], D4○C16, C2×Q32 [×3], C4○D16 [×3], Q32⋊C2 [×6], Q8○D8 [×2], D4○Q32
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D4○Q32
►On 64 points
(1 29 9 21)(2 30 10 22)(3 31 11 23)(4 32 12 24)(5 17 13 25)(6 18 14 26)(7 19 15 27)(8 20 16 28)(33 59 41 51)(34 60 42 52)(35 61 43 53)(36 62 44 54)(37 63 45 55)(38 64 46 56)(39 49 47 57)(40 50 48 58)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 49)(16 50)(17 45)(18 46)(19 47)(20 48)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)
(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 30 9 22)(2 29 10 21)(3 28 11 20)(4 27 12 19)(5 26 13 18)(6 25 14 17)(7 24 15 32)(8 23 16 31)(33 52 41 60)(34 51 42 59)(35 50 43 58)(36 49 44 57)(37 64 45 56)(38 63 46 55)(39 62 47 54)(40 61 48 53)
G:=sub<Sym(64)| (1,29,9,21)(2,30,10,22)(3,31,11,23)(4,32,12,24)(5,17,13,25)(6,18,14,26)(7,19,15,27)(8,20,16,28)(33,59,41,51)(34,60,42,52)(35,61,43,53)(36,62,44,54)(37,63,45,55)(38,64,46,56)(39,49,47,57)(40,50,48,58), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,49)(16,50)(17,45)(18,46)(19,47)(20,48)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44), (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), (1,30,9,22)(2,29,10,21)(3,28,11,20)(4,27,12,19)(5,26,13,18)(6,25,14,17)(7,24,15,32)(8,23,16,31)(33,52,41,60)(34,51,42,59)(35,50,43,58)(36,49,44,57)(37,64,45,56)(38,63,46,55)(39,62,47,54)(40,61,48,53)>;
G:=Group( (1,29,9,21)(2,30,10,22)(3,31,11,23)(4,32,12,24)(5,17,13,25)(6,18,14,26)(7,19,15,27)(8,20,16,28)(33,59,41,51)(34,60,42,52)(35,61,43,53)(36,62,44,54)(37,63,45,55)(38,64,46,56)(39,49,47,57)(40,50,48,58), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,49)(16,50)(17,45)(18,46)(19,47)(20,48)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44), (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), (1,30,9,22)(2,29,10,21)(3,28,11,20)(4,27,12,19)(5,26,13,18)(6,25,14,17)(7,24,15,32)(8,23,16,31)(33,52,41,60)(34,51,42,59)(35,50,43,58)(36,49,44,57)(37,64,45,56)(38,63,46,55)(39,62,47,54)(40,61,48,53) );
G=PermutationGroup([(1,29,9,21),(2,30,10,22),(3,31,11,23),(4,32,12,24),(5,17,13,25),(6,18,14,26),(7,19,15,27),(8,20,16,28),(33,59,41,51),(34,60,42,52),(35,61,43,53),(36,62,44,54),(37,63,45,55),(38,64,46,56),(39,49,47,57),(40,50,48,58)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,49),(16,50),(17,45),(18,46),(19,47),(20,48),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44)], [(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)], [(1,30,9,22),(2,29,10,21),(3,28,11,20),(4,27,12,19),(5,26,13,18),(6,25,14,17),(7,24,15,32),(8,23,16,31),(33,52,41,60),(34,51,42,59),(35,50,43,58),(36,49,44,57),(37,64,45,56),(38,63,46,55),(39,62,47,54),(40,61,48,53)])
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 | D4○Q32 |
| kernel | D4○Q32 | 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 D4○Q32 ►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] >;
D4○Q32 in GAP, Magma, Sage, TeX
D_4\circ Q_{32} % in TeX
G:=Group("D4oQ32"); // 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