metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊6D10, SD16⋊4D10, D20.42D4, C40.3C23, C20.22C24, M4(2)⋊10D10, Dic10.42D4, Dic20⋊2C22, D20.15C23, Dic10.15C23, C4○D4⋊4D10, C8⋊C22⋊5D5, C5⋊D4.5D4, D8⋊3D5⋊2C2, D8⋊D5⋊4C2, (D5×SD16)⋊2C2, C4.116(D4×D5), C5⋊4(D4○SD16), (C8×D5)⋊4C22, (C5×D8)⋊4C22, D4⋊6D10⋊8C2, (Q8×D5)⋊3C22, C8.3(C22×D5), D4⋊D5⋊15C22, D10.56(C2×D4), C8.D10⋊2C2, C20.243(C2×D4), SD16⋊D5⋊2C2, C8⋊D5⋊4C22, C40⋊C2⋊4C22, Q8⋊D5⋊14C22, (D4×D5).3C22, C22.15(D4×D5), C4.22(C23×D5), D20.2C4⋊2C2, D4.8D10⋊4C2, (C2×D4).117D10, D4⋊2D5⋊4C22, C5⋊2C8.26C23, D4.D5⋊14C22, Dic5.62(C2×D4), (C5×SD16)⋊4C22, (C4×D5).14C23, D4.15(C22×D5), (C5×D4).15C23, C5⋊Q16⋊13C22, D4.10D10⋊7C2, Q8.15(C22×D5), (C5×Q8).15C23, (C2×C20).113C23, C4○D20.29C22, C10.123(C22×D4), (C5×M4(2))⋊4C22, (C2×Dic10)⋊40C22, (D4×C10).168C22, C2.96(C2×D4×D5), (C5×C8⋊C22)⋊4C2, (C2×D4.D5)⋊29C2, (C2×C10).68(C2×D4), (C5×C4○D4)⋊7C22, (C2×C5⋊2C8)⋊18C22, (C2×C4).97(C22×D5), SmallGroup(320,1447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊6D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 998 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4○SD16, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C5⋊2C8, D4⋊D5, D4.D5, D4.D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, C2×C5⋊D4, D4×C10, C5×C4○D4, D20.2C4, C8.D10, D8⋊D5, D8⋊3D5, D5×SD16, SD16⋊D5, C2×D4.D5, D4.8D10, C5×C8⋊C22, D4⋊6D10, D4.10D10, D8⋊6D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○SD16, D4×D5, C23×D5, C2×D4×D5, D8⋊6D10
►On 80 points
(1 41 40 78 63 17 55 27)(2 79 56 42 64 28 31 18)(3 43 32 80 65 19 57 29)(4 71 58 44 66 30 33 20)(5 45 34 72 67 11 59 21)(6 73 60 46 68 22 35 12)(7 47 36 74 69 13 51 23)(8 75 52 48 70 24 37 14)(9 49 38 76 61 15 53 25)(10 77 54 50 62 26 39 16)
(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 71)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(31 56)(32 57)(33 58)(34 59)(35 60)(36 51)(37 52)(38 53)(39 54)(40 55)
(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 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 60)(11 17)(12 16)(13 15)(18 20)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 80)(30 79)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(41 45)(42 44)(46 50)(47 49)
G:=sub<Sym(80)| (1,41,40,78,63,17,55,27)(2,79,56,42,64,28,31,18)(3,43,32,80,65,19,57,29)(4,71,58,44,66,30,33,20)(5,45,34,72,67,11,59,21)(6,73,60,46,68,22,35,12)(7,47,36,74,69,13,51,23)(8,75,52,48,70,24,37,14)(9,49,38,76,61,15,53,25)(10,77,54,50,62,26,39,16), (11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,71)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,56)(32,57)(33,58)(34,59)(35,60)(36,51)(37,52)(38,53)(39,54)(40,55), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,60)(11,17)(12,16)(13,15)(18,20)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,80)(30,79)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(41,45)(42,44)(46,50)(47,49)>;
G:=Group( (1,41,40,78,63,17,55,27)(2,79,56,42,64,28,31,18)(3,43,32,80,65,19,57,29)(4,71,58,44,66,30,33,20)(5,45,34,72,67,11,59,21)(6,73,60,46,68,22,35,12)(7,47,36,74,69,13,51,23)(8,75,52,48,70,24,37,14)(9,49,38,76,61,15,53,25)(10,77,54,50,62,26,39,16), (11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,71)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,56)(32,57)(33,58)(34,59)(35,60)(36,51)(37,52)(38,53)(39,54)(40,55), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,60)(11,17)(12,16)(13,15)(18,20)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,80)(30,79)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(41,45)(42,44)(46,50)(47,49) );
G=PermutationGroup([[(1,41,40,78,63,17,55,27),(2,79,56,42,64,28,31,18),(3,43,32,80,65,19,57,29),(4,71,58,44,66,30,33,20),(5,45,34,72,67,11,59,21),(6,73,60,46,68,22,35,12),(7,47,36,74,69,13,51,23),(8,75,52,48,70,24,37,14),(9,49,38,76,61,15,53,25),(10,77,54,50,62,26,39,16)], [(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,71),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,41),(28,42),(29,43),(30,44),(31,56),(32,57),(33,58),(34,59),(35,60),(36,51),(37,52),(38,53),(39,54),(40,55)], [(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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,60),(11,17),(12,16),(13,15),(18,20),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,80),(30,79),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,70),(38,69),(39,68),(40,67),(41,45),(42,44),(46,50),(47,49)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | ··· | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D4○SD16 | D4×D5 | D4×D5 | D8⋊6D10 |
| kernel | D8⋊6D10 | D20.2C4 | C8.D10 | D8⋊D5 | D8⋊3D5 | D5×SD16 | SD16⋊D5 | C2×D4.D5 | D4.8D10 | C5×C8⋊C22 | D4⋊6D10 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of D8⋊6D10 ►in GL8(𝔽41)
| 1 | 0 | 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 40 | 0 | 0 | 0 | 0 | 0 |
| 38 | 3 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 | 26 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 | 15 | 26 |
| 0 | 0 | 0 | 0 | 0 | 30 | 0 | 15 |
| 0 | 0 | 0 | 0 | 11 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 40 | 0 | 0 | 0 | 0 | 0 |
| 38 | 3 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 18 | 0 | 35 | 0 | 0 | 0 | 0 |
| 20 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 38 | 7 | 35 | 0 | 0 | 0 | 0 |
| 20 | 38 | 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 1 |
G:=sub<GL(8,GF(41))| [1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,13,13,40,0,0,0,0,0,28,0,0,40,0,0,0,0,0,0,0,0,15,15,0,11,0,0,0,0,26,26,30,30,0,0,0,0,26,15,0,0,0,0,0,0,0,26,15,0],[1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40],[1,35,23,20,0,0,0,0,6,6,18,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,40,39,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40],[1,35,23,20,0,0,0,0,0,40,38,38,0,0,0,0,0,0,7,8,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,40,39,0,0,0,0,0,0,0,1] >;
D8⋊6D10 in GAP, Magma, Sage, TeX
D_8\rtimes_6D_{10} % in TeX
G:=Group("D8:6D10"); // GroupNames label
G:=SmallGroup(320,1447);
// by ID
G=gap.SmallGroup(320,1447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations