metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2C8, Dic5.20D4, C10.3M4(2), (C2×C4).3F5, C5⋊1(C22⋊C8), C10.3(C2×C8), (C2×C20).3C4, C2.4(D5⋊C8), C2.3(C4.F5), (C22×D5).6C4, C2.1(C22⋊F5), C22.11(C2×F5), C10.2(C22⋊C4), (C2×Dic5).51C22, (C2×C5⋊C8)⋊1C2, (C2×C4×D5).8C2, (C2×C10).6(C2×C4), SmallGroup(160,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊C8
G = < a,b,c | a10=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a7b >
Character table of D10⋊C8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | -i | i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | i | -i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | 1 | ζ87 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ83 | -1 | -1 | 1 | i | -i | -i | i | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | 1 | ζ83 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ87 | -1 | -1 | 1 | i | -i | -i | i | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | 1 | ζ8 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ85 | -1 | -1 | 1 | -i | i | i | -i | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | 1 | ζ85 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ8 | -1 | -1 | 1 | -i | i | i | -i | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | ζ83 | ζ87 | ζ8 | ζ85 | ζ8 | ζ85 | ζ87 | ζ83 | -1 | -1 | 1 | -i | i | i | -i | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | ζ8 | ζ85 | ζ83 | ζ87 | ζ83 | ζ87 | ζ85 | ζ8 | -1 | -1 | 1 | i | -i | -i | i | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | ζ85 | ζ8 | ζ87 | ζ83 | ζ87 | ζ83 | ζ8 | ζ85 | -1 | -1 | 1 | i | -i | -i | i | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | ζ87 | ζ83 | ζ85 | ζ8 | ζ85 | ζ8 | ζ83 | ζ87 | -1 | -1 | 1 | -i | i | i | -i | linear of order 8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√5 | √5 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √5 | -√5 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | -i | i | complex lifted from D5⋊C8, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | i | -i | complex lifted from D5⋊C8, Schur index 2 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4.F5 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4.F5 |
►On 80 points
Generators in S80
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)
(1 14 46 65 21 80 40 60)(2 11 45 68 22 77 39 53)(3 18 44 61 23 74 38 56)(4 15 43 64 24 71 37 59)(5 12 42 67 25 78 36 52)(6 19 41 70 26 75 35 55)(7 16 50 63 27 72 34 58)(8 13 49 66 28 79 33 51)(9 20 48 69 29 76 32 54)(10 17 47 62 30 73 31 57)
G:=sub<Sym(80)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69), (1,14,46,65,21,80,40,60)(2,11,45,68,22,77,39,53)(3,18,44,61,23,74,38,56)(4,15,43,64,24,71,37,59)(5,12,42,67,25,78,36,52)(6,19,41,70,26,75,35,55)(7,16,50,63,27,72,34,58)(8,13,49,66,28,79,33,51)(9,20,48,69,29,76,32,54)(10,17,47,62,30,73,31,57)>;
G:=Group( (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69), (1,14,46,65,21,80,40,60)(2,11,45,68,22,77,39,53)(3,18,44,61,23,74,38,56)(4,15,43,64,24,71,37,59)(5,12,42,67,25,78,36,52)(6,19,41,70,26,75,35,55)(7,16,50,63,27,72,34,58)(8,13,49,66,28,79,33,51)(9,20,48,69,29,76,32,54)(10,17,47,62,30,73,31,57) );
G=PermutationGroup([[(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,80),(19,79),(20,78),(31,44),(32,43),(33,42),(34,41),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,70),(60,69)], [(1,14,46,65,21,80,40,60),(2,11,45,68,22,77,39,53),(3,18,44,61,23,74,38,56),(4,15,43,64,24,71,37,59),(5,12,42,67,25,78,36,52),(6,19,41,70,26,75,35,55),(7,16,50,63,27,72,34,58),(8,13,49,66,28,79,33,51),(9,20,48,69,29,76,32,54),(10,17,47,62,30,73,31,57)]])
D10⋊C8 is a maximal subgroup of
D10.1D8 D10.1Q16 D10.SD16 D10.Q16 C42.6F5 C42.12F5 C42.15F5 C42.7F5 C5⋊C8⋊8D4 C5⋊C8⋊D4 D10⋊M4(2) Dic5⋊M4(2) D20⋊2C8 D10⋊2M4(2) C20⋊M4(2) C4⋊C4.7F5 D10.11M4(2) D10⋊9M4(2) D10⋊10M4(2) (C2×D4).7F5 (C2×Q8).5F5 D30⋊C8 C30.7M4(2)
D10⋊C8 is a maximal quotient of
D20⋊C8 Dic10⋊1C8 D10⋊C16 C20.10M4(2) D20.C8 (C22×C4).F5 C5⋊(C23⋊C8) C10.(C4⋊C8) D30⋊C8 C30.7M4(2)
Matrix representation of D10⋊C8 ►in GL8(𝔽41)
| 40 | 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 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 27 | 23 | 0 |
| 0 | 0 | 0 | 0 | 37 | 27 | 0 | 14 |
| 0 | 0 | 0 | 0 | 14 | 0 | 27 | 37 |
| 0 | 0 | 0 | 0 | 0 | 23 | 27 | 14 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,14,37,14,0,0,0,0,0,27,27,0,23,0,0,0,0,23,0,27,27,0,0,0,0,0,14,37,14] >;
D10⋊C8 in GAP, Magma, Sage, TeX
D_{10}\rtimes C_8 % in TeX
G:=Group("D10:C8"); // GroupNames label
G:=SmallGroup(160,78);
// by ID
G=gap.SmallGroup(160,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,86,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^10=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^7*b>;
// generators/relations
Export
Subgroup lattice of D10⋊C8 in TeX
Character table of D10⋊C8 in TeX