metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1F5, C40.1C4, D10.2Q8, Dic5.11D4, C5⋊2C8.4C4, (C8×D5).4C2, C2.7(C4⋊F5), C10.4(C4⋊C4), C4.11(C2×F5), C5⋊2(C8.C4), C4.F5.2C2, C20.11(C2×C4), (C4×D5).28C22, SmallGroup(160,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.Q8
G = < a,b,c,d | a10=b2=1, c4=a5, d2=a-1bc2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a7b, dcd-1=a5c3 >
Character table of D10.Q8
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10 | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 2 | 5 | 5 | 4 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 20 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 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 | -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 | -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 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | -√2 | √2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C8.C4 |
| ρ12 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | -√2 | √2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C8.C4 |
| ρ13 | 2 | -2 | 0 | 0 | 2i | -2i | 2 | √2 | -√2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C8.C4 |
| ρ14 | 2 | -2 | 0 | 0 | -2i | 2i | 2 | √2 | -√2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C8.C4 |
| ρ15 | 4 | 4 | 0 | 4 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ16 | 4 | 4 | 0 | 4 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ17 | 4 | 4 | 0 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4⋊F5 |
| ρ18 | 4 | 4 | 0 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-5 | -√-5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -√-5 | √-5 | ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 | ζ83ζ54+ζ83ζ5+ζ83+ζ8ζ54+ζ8ζ5 | ζ83ζ53+ζ83ζ52+ζ83+ζ8ζ53+ζ8ζ52 | ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5+ζ8 | complex faithful |
►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 14)(12 13)(15 20)(16 19)(17 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 34)(32 33)(35 40)(36 39)(37 38)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 33 13 28 6 38 18 23)(2 34 14 29 7 39 19 24)(3 35 15 30 8 40 20 25)(4 36 16 21 9 31 11 26)(5 37 17 22 10 32 12 27)(41 61 56 76 46 66 51 71)(42 62 57 77 47 67 52 72)(43 63 58 78 48 68 53 73)(44 64 59 79 49 69 54 74)(45 65 60 80 50 70 55 75)
(1 59 13 44 6 54 18 49)(2 56 12 47 7 51 17 42)(3 53 11 50 8 58 16 45)(4 60 20 43 9 55 15 48)(5 57 19 46 10 52 14 41)(21 75 35 68 26 80 40 63)(22 72 34 61 27 77 39 66)(23 79 33 64 28 74 38 69)(24 76 32 67 29 71 37 62)(25 73 31 70 30 78 36 65)
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)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,33,13,28,6,38,18,23)(2,34,14,29,7,39,19,24)(3,35,15,30,8,40,20,25)(4,36,16,21,9,31,11,26)(5,37,17,22,10,32,12,27)(41,61,56,76,46,66,51,71)(42,62,57,77,47,67,52,72)(43,63,58,78,48,68,53,73)(44,64,59,79,49,69,54,74)(45,65,60,80,50,70,55,75), (1,59,13,44,6,54,18,49)(2,56,12,47,7,51,17,42)(3,53,11,50,8,58,16,45)(4,60,20,43,9,55,15,48)(5,57,19,46,10,52,14,41)(21,75,35,68,26,80,40,63)(22,72,34,61,27,77,39,66)(23,79,33,64,28,74,38,69)(24,76,32,67,29,71,37,62)(25,73,31,70,30,78,36,65)>;
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)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,33,13,28,6,38,18,23)(2,34,14,29,7,39,19,24)(3,35,15,30,8,40,20,25)(4,36,16,21,9,31,11,26)(5,37,17,22,10,32,12,27)(41,61,56,76,46,66,51,71)(42,62,57,77,47,67,52,72)(43,63,58,78,48,68,53,73)(44,64,59,79,49,69,54,74)(45,65,60,80,50,70,55,75), (1,59,13,44,6,54,18,49)(2,56,12,47,7,51,17,42)(3,53,11,50,8,58,16,45)(4,60,20,43,9,55,15,48)(5,57,19,46,10,52,14,41)(21,75,35,68,26,80,40,63)(22,72,34,61,27,77,39,66)(23,79,33,64,28,74,38,69)(24,76,32,67,29,71,37,62)(25,73,31,70,30,78,36,65) );
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),(11,14),(12,13),(15,20),(16,19),(17,18),(21,24),(22,23),(25,30),(26,29),(27,28),(31,34),(32,33),(35,40),(36,39),(37,38),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)], [(1,33,13,28,6,38,18,23),(2,34,14,29,7,39,19,24),(3,35,15,30,8,40,20,25),(4,36,16,21,9,31,11,26),(5,37,17,22,10,32,12,27),(41,61,56,76,46,66,51,71),(42,62,57,77,47,67,52,72),(43,63,58,78,48,68,53,73),(44,64,59,79,49,69,54,74),(45,65,60,80,50,70,55,75)], [(1,59,13,44,6,54,18,49),(2,56,12,47,7,51,17,42),(3,53,11,50,8,58,16,45),(4,60,20,43,9,55,15,48),(5,57,19,46,10,52,14,41),(21,75,35,68,26,80,40,63),(22,72,34,61,27,77,39,66),(23,79,33,64,28,74,38,69),(24,76,32,67,29,71,37,62),(25,73,31,70,30,78,36,65)]])
D10.Q8 is a maximal subgroup of
C16.F5 C80.2C4 D8.F5 Q16.F5 (C8×D5).C4 M4(2).1F5 D8⋊5F5 SD16⋊2F5 Q16⋊5F5 D10.2Dic6 C24.1F5
D10.Q8 is a maximal quotient of
C40⋊1C8 C20.26M4(2) C20.10C42 D10.2Dic6 C24.1F5
Matrix representation of D10.Q8 ►in GL4(𝔽7) generated by
| 6 | 0 | 4 | 2 |
| 5 | 1 | 0 | 0 |
| 6 | 3 | 3 | 4 |
| 3 | 5 | 1 | 5 |
| 6 | 0 | 4 | 2 |
| 0 | 6 | 1 | 4 |
| 0 | 3 | 3 | 1 |
| 0 | 1 | 3 | 6 |
| 6 | 3 | 0 | 2 |
| 1 | 6 | 1 | 1 |
| 2 | 2 | 2 | 3 |
| 3 | 4 | 2 | 6 |
| 1 | 1 | 6 | 1 |
| 0 | 2 | 6 | 4 |
| 3 | 5 | 4 | 6 |
| 5 | 4 | 6 | 0 |
G:=sub<GL(4,GF(7))| [6,5,6,3,0,1,3,5,4,0,3,1,2,0,4,5],[6,0,0,0,0,6,3,1,4,1,3,3,2,4,1,6],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[1,0,3,5,1,2,5,4,6,6,4,6,1,4,6,0] >;
D10.Q8 in GAP, Magma, Sage, TeX
D_{10}.Q_8 % in TeX
G:=Group("D10.Q8"); // GroupNames label
G:=SmallGroup(160,71);
// by ID
G=gap.SmallGroup(160,71);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,151,86,579,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=1,c^4=a^5,d^2=a^-1*b*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^5*c^3>;
// generators/relations
Export
Subgroup lattice of D10.Q8 in TeX
Character table of D10.Q8 in TeX