metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.1D8, D10.3SD16, C4⋊C4⋊1F5, C10.7C4≀C2, (C2×D20)⋊5C4, D10⋊C8⋊1C2, C4⋊D20.1C2, C10.7(C23⋊C4), C2.4(D20⋊C4), D10.3Q8⋊1C2, C2.4(Q8⋊2F5), (C2×Dic5).92D4, (C22×D5).54D4, C5⋊1(C22.SD16), C10.1(D4⋊C4), C22.55(C22⋊F5), C2.10(D10.D4), (C5×C4⋊C4)⋊1C4, (C2×C4).8(C2×F5), (C2×C20).5(C2×C4), (C2×C4×D5).1C22, (C2×C10).18(C22⋊C4), SmallGroup(320,206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.1D8
G = < a,b,c,d | a10=b2=c8=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc-1 >
Subgroups: 570 in 90 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C2.C42, C22⋊C8, C4⋊D4, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C22×D5, C22.SD16, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×C4×D5, C2×D20, C2×D20, C22×F5, D10⋊C8, D10.3Q8, C4⋊D20, D10.1D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, C23⋊C4, D4⋊C4, C4≀C2, C2×F5, C22.SD16, C22⋊F5, D10.D4, D20⋊C4, Q8⋊2F5, D10.1D8
Character table of D10.1D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 20E | 20F | |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 40 | 4 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | -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 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | i | -i | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | -i | i | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -1-i | 1-i | -1+i | 1+i | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 1+i | -1+i | 1-i | -1-i | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ15 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | -√-2 | √-2 | √-2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -1+i | 1+i | -1-i | 1-i | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 1-i | -1-i | 1+i | -1+i | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | √-2 | -√-2 | -√-2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ19 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ21 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | -√5 | √5 | 1 | 1 | √5 | orthogonal lifted from C22⋊F5 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | √5 | -√5 | 1 | 1 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | -√5 | √5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | orthogonal lifted from D10.D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | √5 | -√5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | orthogonal lifted from D10.D4 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | -√5 | √5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | orthogonal lifted from D10.D4 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | √5 | -√5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | orthogonal lifted from D10.D4 |
| ρ28 | 8 | 8 | -8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2F5 |
| ρ29 | 8 | -8 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D20⋊C4, Schur index 2 |
►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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 28)(22 27)(23 26)(24 25)(29 30)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 53)(54 60)(55 59)(56 58)(61 63)(64 70)(65 69)(66 68)(71 73)(74 80)(75 79)(76 78)
(1 65 35 75 30 55 45 11)(2 62 34 78 21 52 44 14)(3 69 33 71 22 59 43 17)(4 66 32 74 23 56 42 20)(5 63 31 77 24 53 41 13)(6 70 40 80 25 60 50 16)(7 67 39 73 26 57 49 19)(8 64 38 76 27 54 48 12)(9 61 37 79 28 51 47 15)(10 68 36 72 29 58 46 18)
(1 30)(2 27 10 23)(3 24 9 26)(4 21 8 29)(5 28 7 22)(6 25)(11 60 16 55)(12 57 15 58)(13 54 14 51)(17 52 20 53)(18 59 19 56)(31 32 39 38)(33 36 37 34)(35 40)(41 42 49 48)(43 46 47 44)(45 50)(61 77 64 78)(62 74 63 71)(65 75 70 80)(66 72 69 73)(67 79 68 76)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,53)(54,60)(55,59)(56,58)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78), (1,65,35,75,30,55,45,11)(2,62,34,78,21,52,44,14)(3,69,33,71,22,59,43,17)(4,66,32,74,23,56,42,20)(5,63,31,77,24,53,41,13)(6,70,40,80,25,60,50,16)(7,67,39,73,26,57,49,19)(8,64,38,76,27,54,48,12)(9,61,37,79,28,51,47,15)(10,68,36,72,29,58,46,18), (1,30)(2,27,10,23)(3,24,9,26)(4,21,8,29)(5,28,7,22)(6,25)(11,60,16,55)(12,57,15,58)(13,54,14,51)(17,52,20,53)(18,59,19,56)(31,32,39,38)(33,36,37,34)(35,40)(41,42,49,48)(43,46,47,44)(45,50)(61,77,64,78)(62,74,63,71)(65,75,70,80)(66,72,69,73)(67,79,68,76)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,53)(54,60)(55,59)(56,58)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78), (1,65,35,75,30,55,45,11)(2,62,34,78,21,52,44,14)(3,69,33,71,22,59,43,17)(4,66,32,74,23,56,42,20)(5,63,31,77,24,53,41,13)(6,70,40,80,25,60,50,16)(7,67,39,73,26,57,49,19)(8,64,38,76,27,54,48,12)(9,61,37,79,28,51,47,15)(10,68,36,72,29,58,46,18), (1,30)(2,27,10,23)(3,24,9,26)(4,21,8,29)(5,28,7,22)(6,25)(11,60,16,55)(12,57,15,58)(13,54,14,51)(17,52,20,53)(18,59,19,56)(31,32,39,38)(33,36,37,34)(35,40)(41,42,49,48)(43,46,47,44)(45,50)(61,77,64,78)(62,74,63,71)(65,75,70,80)(66,72,69,73)(67,79,68,76) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,28),(22,27),(23,26),(24,25),(29,30),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,53),(54,60),(55,59),(56,58),(61,63),(64,70),(65,69),(66,68),(71,73),(74,80),(75,79),(76,78)], [(1,65,35,75,30,55,45,11),(2,62,34,78,21,52,44,14),(3,69,33,71,22,59,43,17),(4,66,32,74,23,56,42,20),(5,63,31,77,24,53,41,13),(6,70,40,80,25,60,50,16),(7,67,39,73,26,57,49,19),(8,64,38,76,27,54,48,12),(9,61,37,79,28,51,47,15),(10,68,36,72,29,58,46,18)], [(1,30),(2,27,10,23),(3,24,9,26),(4,21,8,29),(5,28,7,22),(6,25),(11,60,16,55),(12,57,15,58),(13,54,14,51),(17,52,20,53),(18,59,19,56),(31,32,39,38),(33,36,37,34),(35,40),(41,42,49,48),(43,46,47,44),(45,50),(61,77,64,78),(62,74,63,71),(65,75,70,80),(66,72,69,73),(67,79,68,76)]])
Matrix representation of D10.1D8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 26 | 15 | 0 | 0 | 0 | 0 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 4 | 6 |
| 0 | 0 | 38 | 40 | 8 | 40 |
| 0 | 0 | 1 | 33 | 1 | 3 |
| 0 | 0 | 35 | 37 | 7 | 7 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 38 | 19 | 0 |
| 0 | 0 | 22 | 38 | 0 | 3 |
| 0 | 0 | 3 | 0 | 38 | 22 |
| 0 | 0 | 0 | 19 | 38 | 3 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40,0,0,0,0,0,1],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,34,38,1,35,0,0,34,40,33,37,0,0,4,8,1,7,0,0,6,40,3,7],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,3,22,3,0,0,0,38,38,0,19,0,0,19,0,38,38,0,0,0,3,22,3] >;
D10.1D8 in GAP, Magma, Sage, TeX
D_{10}._1D_8 % in TeX
G:=Group("D10.1D8"); // GroupNames label
G:=SmallGroup(320,206);
// by ID
G=gap.SmallGroup(320,206);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,1571,570,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^-1>;
// generators/relations
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