metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊15D10, Q16⋊13D10, D20.45D4, SD16⋊11D10, D40⋊19C22, C20.16C24, C40.38C23, Dic10.45D4, D20.11C23, C4○D8⋊4D5, (D5×D8)⋊7C2, C5⋊3(D4○D8), C4○D4⋊1D10, (C2×C8)⋊13D10, C5⋊D4.1D4, (C2×D40)⋊23C2, D40⋊C2⋊6C2, D4⋊D5⋊3C22, C4.143(D4×D5), (D4×D5)⋊2C22, (C8×D5)⋊8C22, Q8⋊D5⋊2C22, D4⋊D10⋊7C2, D4⋊8D10⋊5C2, Q8.D10⋊7C2, C22.8(D4×D5), (C2×C40)⋊12C22, D10.52(C2×D4), C20.349(C2×D4), (C5×D8)⋊13C22, C5⋊2C8.7C23, (C4×D5).9C23, C8.16(C22×D5), C4.16(C23×D5), D20.3C4⋊7C2, (C2×D20)⋊34C22, C8⋊D5⋊11C22, Dic5.58(C2×D4), Q8⋊2D5⋊2C22, (C5×Q16)⋊11C22, D4.10(C22×D5), (C5×D4).10C23, Q8.10(C22×D5), (C5×Q8).10C23, (C2×C20).533C23, C4○D20.54C22, (C5×SD16)⋊11C22, C10.117(C22×D4), C4.Dic5⋊30C22, C2.90(C2×D4×D5), (C5×C4○D8)⋊5C2, (C2×C10).13(C2×D4), (C5×C4○D4)⋊3C22, (C2×C4).232(C22×D5), SmallGroup(320,1441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊15D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >
Subgroups: 1238 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×D8, C4○D8, C4○D8, C8⋊C22, 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, D20, D20, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4○D8, C8×D5, C8⋊D5, D40, C4.Dic5, D4⋊D5, Q8⋊D5, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, Q8⋊2D5, C5×C4○D4, D20.3C4, C2×D40, D5×D8, D40⋊C2, Q8.D10, D4⋊D10, C5×C4○D8, D4⋊8D10, D8⋊15D10
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⋊15D10
►On 80 points
(1 19 30 46 63 73 40 55)(2 20 21 47 64 74 31 56)(3 11 22 48 65 75 32 57)(4 12 23 49 66 76 33 58)(5 13 24 50 67 77 34 59)(6 14 25 41 68 78 35 60)(7 15 26 42 69 79 36 51)(8 16 27 43 70 80 37 52)(9 17 28 44 61 71 38 53)(10 18 29 45 62 72 39 54)
(2 64)(4 66)(6 68)(8 70)(10 62)(11 57)(12 49)(13 59)(14 41)(15 51)(16 43)(17 53)(18 45)(19 55)(20 47)(22 32)(24 34)(26 36)(28 38)(30 40)(42 79)(44 71)(46 73)(48 75)(50 77)(52 80)(54 72)(56 74)(58 76)(60 78)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 20)(10 19)(21 53)(22 52)(23 51)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(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)| (1,19,30,46,63,73,40,55)(2,20,21,47,64,74,31,56)(3,11,22,48,65,75,32,57)(4,12,23,49,66,76,33,58)(5,13,24,50,67,77,34,59)(6,14,25,41,68,78,35,60)(7,15,26,42,69,79,36,51)(8,16,27,43,70,80,37,52)(9,17,28,44,61,71,38,53)(10,18,29,45,62,72,39,54), (2,64)(4,66)(6,68)(8,70)(10,62)(11,57)(12,49)(13,59)(14,41)(15,51)(16,43)(17,53)(18,45)(19,55)(20,47)(22,32)(24,34)(26,36)(28,38)(30,40)(42,79)(44,71)(46,73)(48,75)(50,77)(52,80)(54,72)(56,74)(58,76)(60,78), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,20)(10,19)(21,53)(22,52)(23,51)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(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( (1,19,30,46,63,73,40,55)(2,20,21,47,64,74,31,56)(3,11,22,48,65,75,32,57)(4,12,23,49,66,76,33,58)(5,13,24,50,67,77,34,59)(6,14,25,41,68,78,35,60)(7,15,26,42,69,79,36,51)(8,16,27,43,70,80,37,52)(9,17,28,44,61,71,38,53)(10,18,29,45,62,72,39,54), (2,64)(4,66)(6,68)(8,70)(10,62)(11,57)(12,49)(13,59)(14,41)(15,51)(16,43)(17,53)(18,45)(19,55)(20,47)(22,32)(24,34)(26,36)(28,38)(30,40)(42,79)(44,71)(46,73)(48,75)(50,77)(52,80)(54,72)(56,74)(58,76)(60,78), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,20)(10,19)(21,53)(22,52)(23,51)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(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([[(1,19,30,46,63,73,40,55),(2,20,21,47,64,74,31,56),(3,11,22,48,65,75,32,57),(4,12,23,49,66,76,33,58),(5,13,24,50,67,77,34,59),(6,14,25,41,68,78,35,60),(7,15,26,42,69,79,36,51),(8,16,27,43,70,80,37,52),(9,17,28,44,61,71,38,53),(10,18,29,45,62,72,39,54)], [(2,64),(4,66),(6,68),(8,70),(10,62),(11,57),(12,49),(13,59),(14,41),(15,51),(16,43),(17,53),(18,45),(19,55),(20,47),(22,32),(24,34),(26,36),(28,38),(30,40),(42,79),(44,71),(46,73),(48,75),(50,77),(52,80),(54,72),(56,74),(58,76),(60,78)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,20),(10,19),(21,53),(22,52),(23,51),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(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)]])
50 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 | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| 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 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D4○D8 | D4×D5 | D4×D5 | D8⋊15D10 |
| kernel | D8⋊15D10 | D20.3C4 | C2×D40 | D5×D8 | D40⋊C2 | Q8.D10 | D4⋊D10 | C5×C4○D8 | D4⋊8D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D8⋊15D10 ►in GL4(𝔽41) generated by
| 29 | 0 | 29 | 0 |
| 0 | 29 | 0 | 29 |
| 12 | 0 | 29 | 0 |
| 0 | 12 | 0 | 29 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 27 |
| 0 | 0 | 14 | 14 |
| 11 | 14 | 0 | 0 |
| 27 | 27 | 0 | 0 |
| 9 | 15 | 32 | 26 |
| 37 | 32 | 4 | 9 |
| 32 | 26 | 32 | 26 |
| 4 | 9 | 4 | 9 |
G:=sub<GL(4,GF(41))| [29,0,12,0,0,29,0,12,29,0,29,0,0,29,0,29],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,11,27,0,0,14,27,30,14,0,0,27,14,0,0],[9,37,32,4,15,32,26,9,32,4,32,4,26,9,26,9] >;
D8⋊15D10 in GAP, Magma, Sage, TeX
D_8\rtimes_{15}D_{10} % in TeX
G:=Group("D8:15D10"); // GroupNames label
G:=SmallGroup(320,1441);
// by ID
G=gap.SmallGroup(320,1441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations