metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊8D10, Q16⋊7D10, C20.51D8, C40.32D4, C40.29C23, D40.11C22, C4○D8⋊2D5, C5⋊D16⋊6C2, (C2×C10).9D8, (C2×D40)⋊22C2, C5⋊5(C16⋊C22), C10.68(C2×D8), (C2×C8).98D10, C5⋊SD32⋊6C2, (C5×D8)⋊8C22, C8.7(C5⋊D4), C20.4C8⋊6C2, C4.24(D4⋊D5), C5⋊2C16⋊4C22, C20.191(C2×D4), (C2×C20).185D4, (C5×Q16)⋊7C22, C8.35(C22×D5), C22.5(D4⋊D5), (C2×C40).104C22, (C5×C4○D8)⋊4C2, C2.23(C2×D4⋊D5), C4.17(C2×C5⋊D4), (C2×C4).81(C5⋊D4), SmallGroup(320,820)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=ab, dcd=c-1 >
Subgroups: 494 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, M5(2), D16, SD32, C2×D8, C4○D8, C40, D20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C16⋊C22, C5⋊2C16, D40, D40, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×D20, C5×C4○D4, C20.4C8, C5⋊D16, C5⋊SD32, C2×D40, C5×C4○D8, D8⋊D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, C16⋊C22, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, D8⋊D10
►On 80 points
Generators in S80
(1 33 13 28 8 40 18 23)(2 34 14 29 9 36 19 24)(3 35 15 30 10 37 20 25)(4 31 11 26 6 38 16 21)(5 32 12 27 7 39 17 22)(41 63 58 78 46 68 53 73)(42 64 59 79 47 69 54 74)(43 65 60 80 48 70 55 75)(44 66 51 71 49 61 56 76)(45 67 52 72 50 62 57 77)
(1 73)(2 79)(3 75)(4 71)(5 77)(6 76)(7 72)(8 78)(9 74)(10 80)(11 66)(12 62)(13 68)(14 64)(15 70)(16 61)(17 67)(18 63)(19 69)(20 65)(21 49)(22 45)(23 41)(24 47)(25 43)(26 44)(27 50)(28 46)(29 42)(30 48)(31 51)(32 57)(33 53)(34 59)(35 55)(36 54)(37 60)(38 56)(39 52)(40 58)
(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 5)(2 4)(6 9)(7 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 34)(22 33)(23 32)(24 31)(25 35)(26 36)(27 40)(28 39)(29 38)(30 37)(41 62)(42 61)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)
G:=sub<Sym(80)| (1,33,13,28,8,40,18,23)(2,34,14,29,9,36,19,24)(3,35,15,30,10,37,20,25)(4,31,11,26,6,38,16,21)(5,32,12,27,7,39,17,22)(41,63,58,78,46,68,53,73)(42,64,59,79,47,69,54,74)(43,65,60,80,48,70,55,75)(44,66,51,71,49,61,56,76)(45,67,52,72,50,62,57,77), (1,73)(2,79)(3,75)(4,71)(5,77)(6,76)(7,72)(8,78)(9,74)(10,80)(11,66)(12,62)(13,68)(14,64)(15,70)(16,61)(17,67)(18,63)(19,69)(20,65)(21,49)(22,45)(23,41)(24,47)(25,43)(26,44)(27,50)(28,46)(29,42)(30,48)(31,51)(32,57)(33,53)(34,59)(35,55)(36,54)(37,60)(38,56)(39,52)(40,58), (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), (1,5)(2,4)(6,9)(7,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,34)(22,33)(23,32)(24,31)(25,35)(26,36)(27,40)(28,39)(29,38)(30,37)(41,62)(42,61)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80)>;
G:=Group( (1,33,13,28,8,40,18,23)(2,34,14,29,9,36,19,24)(3,35,15,30,10,37,20,25)(4,31,11,26,6,38,16,21)(5,32,12,27,7,39,17,22)(41,63,58,78,46,68,53,73)(42,64,59,79,47,69,54,74)(43,65,60,80,48,70,55,75)(44,66,51,71,49,61,56,76)(45,67,52,72,50,62,57,77), (1,73)(2,79)(3,75)(4,71)(5,77)(6,76)(7,72)(8,78)(9,74)(10,80)(11,66)(12,62)(13,68)(14,64)(15,70)(16,61)(17,67)(18,63)(19,69)(20,65)(21,49)(22,45)(23,41)(24,47)(25,43)(26,44)(27,50)(28,46)(29,42)(30,48)(31,51)(32,57)(33,53)(34,59)(35,55)(36,54)(37,60)(38,56)(39,52)(40,58), (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), (1,5)(2,4)(6,9)(7,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,34)(22,33)(23,32)(24,31)(25,35)(26,36)(27,40)(28,39)(29,38)(30,37)(41,62)(42,61)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80) );
G=PermutationGroup([[(1,33,13,28,8,40,18,23),(2,34,14,29,9,36,19,24),(3,35,15,30,10,37,20,25),(4,31,11,26,6,38,16,21),(5,32,12,27,7,39,17,22),(41,63,58,78,46,68,53,73),(42,64,59,79,47,69,54,74),(43,65,60,80,48,70,55,75),(44,66,51,71,49,61,56,76),(45,67,52,72,50,62,57,77)], [(1,73),(2,79),(3,75),(4,71),(5,77),(6,76),(7,72),(8,78),(9,74),(10,80),(11,66),(12,62),(13,68),(14,64),(15,70),(16,61),(17,67),(18,63),(19,69),(20,65),(21,49),(22,45),(23,41),(24,47),(25,43),(26,44),(27,50),(28,46),(29,42),(30,48),(31,51),(32,57),(33,53),(34,59),(35,55),(36,54),(37,60),(38,56),(39,52),(40,58)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,19),(12,18),(13,17),(14,16),(15,20),(21,34),(22,33),(23,32),(24,31),(25,35),(26,36),(27,40),(28,39),(29,38),(30,37),(41,62),(42,61),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,80)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 8 | 40 | 40 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C16⋊C22 | D4⋊D5 | D4⋊D5 | D8⋊D10 |
| kernel | D8⋊D10 | C20.4C8 | C5⋊D16 | C5⋊SD32 | C2×D40 | C5×C4○D8 | C40 | C2×C20 | C4○D8 | C20 | C2×C10 | C2×C8 | D8 | Q16 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D8⋊D10 ►in GL4(𝔽241) generated by
| 20 | 212 | 0 | 0 |
| 29 | 199 | 0 | 0 |
| 0 | 212 | 228 | 29 |
| 29 | 208 | 4 | 232 |
| 29 | 29 | 198 | 170 |
| 20 | 20 | 22 | 192 |
| 0 | 29 | 13 | 212 |
| 20 | 33 | 194 | 179 |
| 240 | 52 | 0 | 0 |
| 189 | 52 | 0 | 0 |
| 175 | 227 | 190 | 189 |
| 131 | 184 | 51 | 0 |
| 1 | 0 | 0 | 0 |
| 52 | 240 | 0 | 0 |
| 9 | 8 | 214 | 20 |
| 15 | 204 | 60 | 27 |
G:=sub<GL(4,GF(241))| [20,29,0,29,212,199,212,208,0,0,228,4,0,0,29,232],[29,20,0,20,29,20,29,33,198,22,13,194,170,192,212,179],[240,189,175,131,52,52,227,184,0,0,190,51,0,0,189,0],[1,52,9,15,0,240,8,204,0,0,214,60,0,0,20,27] >;
D8⋊D10 in GAP, Magma, Sage, TeX
D_8\rtimes D_{10} % in TeX
G:=Group("D8:D10"); // GroupNames label
G:=SmallGroup(320,820);
// by ID
G=gap.SmallGroup(320,820);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,675,185,192,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations