metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.1D4, (C2×C4).F5, (C2×C20).1C4, C5⋊(C4.10D4), C22.F5.C2, C22.3(C2×F5), (C2×Dic5).2C4, C2.5(C22⋊F5), C10.3(C22⋊C4), (C2×Dic10).2C2, (C2×Dic5).23C22, (C2×C10).8(C2×C4), SmallGroup(160,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.D4
G = < a,b,c,d | a10=1, b2=c4=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, cbc-1=a5b, bd=db, dcd-1=bc3 >
Character table of Dic5.D4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 4 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 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 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 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 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √5 | -√5 | -√5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ12 | 4 | 4 | 4 | -4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ13 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√5 | √5 | √5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√5 | √5 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | symplectic faithful, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √5 | -√5 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | √5 | -√5 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ53+2ζ43ζ5+ζ43 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -√5 | √5 | 2ζ4ζ52+2ζ4ζ5+ζ4 | 2ζ43ζ54+2ζ43ζ52+ζ43 | 2ζ43ζ53+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ53+ζ4 | symplectic faithful, 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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 79 66 74)(62 78 67 73)(63 77 68 72)(64 76 69 71)(65 75 70 80)
(1 73 11 62 6 78 16 67)(2 80 20 65 7 75 15 70)(3 77 19 68 8 72 14 63)(4 74 18 61 9 79 13 66)(5 71 17 64 10 76 12 69)(21 57 34 43 26 52 39 48)(22 54 33 46 27 59 38 41)(23 51 32 49 28 56 37 44)(24 58 31 42 29 53 36 47)(25 55 40 45 30 60 35 50)
(1 58 16 42 6 53 11 47)(2 55 15 45 7 60 20 50)(3 52 14 48 8 57 19 43)(4 59 13 41 9 54 18 46)(5 56 12 44 10 51 17 49)(21 77 39 68 26 72 34 63)(22 74 38 61 27 79 33 66)(23 71 37 64 28 76 32 69)(24 78 36 67 29 73 31 62)(25 75 35 70 30 80 40 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), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,79,66,74)(62,78,67,73)(63,77,68,72)(64,76,69,71)(65,75,70,80), (1,73,11,62,6,78,16,67)(2,80,20,65,7,75,15,70)(3,77,19,68,8,72,14,63)(4,74,18,61,9,79,13,66)(5,71,17,64,10,76,12,69)(21,57,34,43,26,52,39,48)(22,54,33,46,27,59,38,41)(23,51,32,49,28,56,37,44)(24,58,31,42,29,53,36,47)(25,55,40,45,30,60,35,50), (1,58,16,42,6,53,11,47)(2,55,15,45,7,60,20,50)(3,52,14,48,8,57,19,43)(4,59,13,41,9,54,18,46)(5,56,12,44,10,51,17,49)(21,77,39,68,26,72,34,63)(22,74,38,61,27,79,33,66)(23,71,37,64,28,76,32,69)(24,78,36,67,29,73,31,62)(25,75,35,70,30,80,40,65)>;
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), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,79,66,74)(62,78,67,73)(63,77,68,72)(64,76,69,71)(65,75,70,80), (1,73,11,62,6,78,16,67)(2,80,20,65,7,75,15,70)(3,77,19,68,8,72,14,63)(4,74,18,61,9,79,13,66)(5,71,17,64,10,76,12,69)(21,57,34,43,26,52,39,48)(22,54,33,46,27,59,38,41)(23,51,32,49,28,56,37,44)(24,58,31,42,29,53,36,47)(25,55,40,45,30,60,35,50), (1,58,16,42,6,53,11,47)(2,55,15,45,7,60,20,50)(3,52,14,48,8,57,19,43)(4,59,13,41,9,54,18,46)(5,56,12,44,10,51,17,49)(21,77,39,68,26,72,34,63)(22,74,38,61,27,79,33,66)(23,71,37,64,28,76,32,69)(24,78,36,67,29,73,31,62)(25,75,35,70,30,80,40,65) );
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)], [(1,16,6,11),(2,15,7,20),(3,14,8,19),(4,13,9,18),(5,12,10,17),(21,39,26,34),(22,38,27,33),(23,37,28,32),(24,36,29,31),(25,35,30,40),(41,54,46,59),(42,53,47,58),(43,52,48,57),(44,51,49,56),(45,60,50,55),(61,79,66,74),(62,78,67,73),(63,77,68,72),(64,76,69,71),(65,75,70,80)], [(1,73,11,62,6,78,16,67),(2,80,20,65,7,75,15,70),(3,77,19,68,8,72,14,63),(4,74,18,61,9,79,13,66),(5,71,17,64,10,76,12,69),(21,57,34,43,26,52,39,48),(22,54,33,46,27,59,38,41),(23,51,32,49,28,56,37,44),(24,58,31,42,29,53,36,47),(25,55,40,45,30,60,35,50)], [(1,58,16,42,6,53,11,47),(2,55,15,45,7,60,20,50),(3,52,14,48,8,57,19,43),(4,59,13,41,9,54,18,46),(5,56,12,44,10,51,17,49),(21,77,39,68,26,72,34,63),(22,74,38,61,27,79,33,66),(23,71,37,64,28,76,32,69),(24,78,36,67,29,73,31,62),(25,75,35,70,30,80,40,65)]])
Dic5.D4 is a maximal subgroup of
C42.F5 C42.2F5 (D4×C10).C4 (Q8×C10).C4 (C4×D5).D4 (C2×D4).9F5 (C2×Q8).7F5 Dic5.4D12 (C2×C60).C4
Dic5.D4 is a maximal quotient of
C10.C4≀C2 Dic5.D8 (C2×C20)⋊1C8 (C22×C4).F5 C22.F5⋊C4 Dic5.4D12 (C2×C60).C4
Matrix representation of Dic5.D4 ►in GL8(𝔽41)
| 34 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 21 | 0 | 40 | 6 | 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 | 40 |
| 15 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 28 | 25 | 16 | 0 | 0 | 0 | 0 |
| 0 | 26 | 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 24 | 1 | 21 |
| 0 | 0 | 0 | 0 | 21 | 25 | 37 | 40 |
| 31 | 35 | 26 | 1 | 0 | 0 | 0 | 0 |
| 27 | 13 | 32 | 14 | 0 | 0 | 0 | 0 |
| 2 | 19 | 37 | 2 | 0 | 0 | 0 | 0 |
| 25 | 37 | 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 37 | 7 | 22 |
| 0 | 0 | 0 | 0 | 30 | 26 | 31 | 21 |
| 0 | 0 | 0 | 0 | 33 | 38 | 24 | 11 |
| 0 | 0 | 0 | 0 | 37 | 17 | 6 | 22 |
| 10 | 6 | 15 | 40 | 0 | 0 | 0 | 0 |
| 14 | 28 | 9 | 27 | 0 | 0 | 0 | 0 |
| 39 | 22 | 4 | 39 | 0 | 0 | 0 | 0 |
| 16 | 4 | 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 4 | 26 | 1 | 21 |
| 0 | 0 | 0 | 0 | 1 | 1 | 39 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 25 | 15 |
G:=sub<GL(8,GF(41))| [34,40,21,21,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[15,3,2,0,0,0,0,0,21,26,28,26,0,0,0,0,0,0,25,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,31,4,21,0,0,0,0,37,0,24,25,0,0,0,0,0,0,1,37,0,0,0,0,0,0,21,40],[31,27,2,25,0,0,0,0,35,13,19,37,0,0,0,0,26,32,37,23,0,0,0,0,1,14,2,1,0,0,0,0,0,0,0,0,10,30,33,37,0,0,0,0,37,26,38,17,0,0,0,0,7,31,24,6,0,0,0,0,22,21,11,22],[10,14,39,16,0,0,0,0,6,28,22,4,0,0,0,0,15,9,4,18,0,0,0,0,40,27,39,40,0,0,0,0,0,0,0,0,2,4,1,0,0,0,0,0,0,26,1,40,0,0,0,0,37,1,39,25,0,0,0,0,0,21,0,15] >;
Dic5.D4 in GAP, Magma, Sage, TeX
{\rm Dic}_5.D_4 % in TeX
G:=Group("Dic5.D4"); // GroupNames label
G:=SmallGroup(160,80);
// by ID
G=gap.SmallGroup(160,80);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,86,579,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^10=1,b^2=c^4=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations
Export
Subgroup lattice of Dic5.D4 in TeX
Character table of Dic5.D4 in TeX