metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊5D10, SD16⋊3D10, D20.41D4, D40⋊2C22, C40.2C23, M4(2)⋊9D10, C20.21C24, Dic10.41D4, D20.14C23, Dic10.14C23, (D5×D8)⋊2C2, C5⋊4(D4○D8), C4○D4⋊3D10, C8⋊C22⋊4D5, C5⋊D4.4D4, D8⋊D5⋊3C2, (C2×D4)⋊15D10, D40⋊C2⋊2C2, C8⋊D10⋊2C2, C4.115(D4×D5), (C8×D5)⋊3C22, (C5×D8)⋊3C22, (D4×D5)⋊3C22, D4⋊8D10⋊7C2, D4⋊6D10⋊7C2, C8.2(C22×D5), D4⋊D5⋊14C22, D10.55(C2×D4), C20.242(C2×D4), C4○D20⋊8C22, C40⋊C2⋊3C22, C8⋊D5⋊3C22, Q8⋊D5⋊13C22, C4.21(C23×D5), C22.14(D4×D5), SD16⋊3D5⋊2C2, D4.8D10⋊3C2, D20.2C4⋊1C2, (C2×D20)⋊36C22, (D4×C10)⋊23C22, D4⋊2D5⋊3C22, C5⋊2C8.25C23, D4.D5⋊13C22, Dic5.61(C2×D4), Q8⋊2D5⋊3C22, (C5×SD16)⋊3C22, (C5×D4).14C23, D4.14(C22×D5), C5⋊Q16⋊12C22, (C4×D5).13C23, Q8.14(C22×D5), (C5×Q8).14C23, (C2×C20).112C23, C10.122(C22×D4), (C5×M4(2))⋊3C22, C2.95(C2×D4×D5), (C2×D4⋊D5)⋊29C2, (C5×C8⋊C22)⋊3C2, (C2×C10).67(C2×D4), (C5×C4○D4)⋊6C22, (C2×C5⋊2C8)⋊17C22, (C2×C4).96(C22×D5), SmallGroup(320,1446)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊5D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 1190 in 268 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, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×D8, C4○D8, C8⋊C22, C8⋊C22, 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4○D8, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C5⋊2C8, D4⋊D5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×D8, D8⋊D5, D40⋊C2, SD16⋊3D5, C2×D4⋊D5, D4.8D10, C5×C8⋊C22, D4⋊6D10, D4⋊8D10, D8⋊5D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○D8, D4×D5, C23×D5, C2×D4×D5, D8⋊5D10
►On 80 points
(1 30 53 46 63 73 40 16)(2 74 54 17 64 21 31 47)(3 22 55 48 65 75 32 18)(4 76 56 19 66 23 33 49)(5 24 57 50 67 77 34 20)(6 78 58 11 68 25 35 41)(7 26 59 42 69 79 36 12)(8 80 60 13 70 27 37 43)(9 28 51 44 61 71 38 14)(10 72 52 15 62 29 39 45)
(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 51)(39 52)(40 53)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 60)(19 59)(20 58)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)
G:=sub<Sym(80)| (1,30,53,46,63,73,40,16)(2,74,54,17,64,21,31,47)(3,22,55,48,65,75,32,18)(4,76,56,19,66,23,33,49)(5,24,57,50,67,77,34,20)(6,78,58,11,68,25,35,41)(7,26,59,42,69,79,36,12)(8,80,60,13,70,27,37,43)(9,28,51,44,61,71,38,14)(10,72,52,15,62,29,39,45), (11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,51)(39,52)(40,53)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,60)(19,59)(20,58)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)>;
G:=Group( (1,30,53,46,63,73,40,16)(2,74,54,17,64,21,31,47)(3,22,55,48,65,75,32,18)(4,76,56,19,66,23,33,49)(5,24,57,50,67,77,34,20)(6,78,58,11,68,25,35,41)(7,26,59,42,69,79,36,12)(8,80,60,13,70,27,37,43)(9,28,51,44,61,71,38,14)(10,72,52,15,62,29,39,45), (11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,21)(18,22)(19,23)(20,24)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,51)(39,52)(40,53)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,57)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,60)(19,59)(20,58)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75) );
G=PermutationGroup([[(1,30,53,46,63,73,40,16),(2,74,54,17,64,21,31,47),(3,22,55,48,65,75,32,18),(4,76,56,19,66,23,33,49),(5,24,57,50,67,77,34,20),(6,78,58,11,68,25,35,41),(7,26,59,42,69,79,36,12),(8,80,60,13,70,27,37,43),(9,28,51,44,61,71,38,14),(10,72,52,15,62,29,39,45)], [(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,21),(18,22),(19,23),(20,24),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,51),(39,52),(40,53),(41,78),(42,79),(43,80),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,57),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,60),(19,59),(20,58),(31,44),(32,43),(33,42),(34,41),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(61,74),(62,73),(63,72),(64,71),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 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 | 20 | 20 | 2 | 2 | 4 | 10 | 10 | 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○D8 | D4×D5 | D4×D5 | D8⋊5D10 |
| kernel | D8⋊5D10 | D20.2C4 | C8⋊D10 | D5×D8 | D8⋊D5 | D40⋊C2 | SD16⋊3D5 | C2×D4⋊D5 | D4.8D10 | C5×C8⋊C22 | D4⋊6D10 | D4⋊8D10 | 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⋊5D10 ►in GL8(𝔽41)
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 21 | 0 | 34 |
| 0 | 0 | 0 | 0 | 3 | 35 | 6 | 24 |
| 1 | 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 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 25 | 40 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 37 | 9 | 0 |
| 0 | 0 | 0 | 0 | 16 | 27 | 32 | 5 |
| 0 | 0 | 0 | 0 | 20 | 6 | 30 | 25 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 18 |
| 0 | 0 | 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 | 0 | 22 |
| 0 | 0 | 0 | 0 | 26 | 3 | 30 | 19 |
| 0 | 0 | 0 | 0 | 29 | 37 | 27 | 5 |
| 0 | 0 | 0 | 0 | 30 | 34 | 26 | 0 |
G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,29,12,18,3,0,0,0,0,29,29,21,35,0,0,0,0,0,0,0,6,0,0,0,0,0,0,34,24],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,1,0,25,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,37,1],[6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,7,16,20,1,0,0,0,0,37,27,6,1,0,0,0,0,9,32,30,0,0,0,0,0,0,5,25,18],[0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,0,0,11,26,29,30,0,0,0,0,11,3,37,34,0,0,0,0,0,30,27,26,0,0,0,0,22,19,5,0] >;
D8⋊5D10 in GAP, Magma, Sage, TeX
D_8\rtimes_5D_{10} % in TeX
G:=Group("D8:5D10"); // GroupNames label
G:=SmallGroup(320,1446);
// by ID
G=gap.SmallGroup(320,1446);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,438,235,102,12550]);
// 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=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations