Copied to
clipboard

## G = D4.D5order 80 = 24·5

### The non-split extension by D4 of D5 acting via D5/C5=C2

Aliases: D4.D5, C52SD16, C4.2D10, C10.8D4, Dic102C2, C20.2C22, C52C82C2, (C5×D4).1C2, C2.5(C5⋊D4), SmallGroup(80,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.D5
 Chief series C1 — C5 — C10 — C20 — Dic10 — D4.D5
 Lower central C5 — C10 — C20 — D4.D5
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D5
G = < a,b,c,d | a4=b2=c5=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Character table of D4.D5

 class 1 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 20A 20B size 1 1 4 2 20 2 2 10 10 2 2 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 -2 0 2 2 0 0 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 2 2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 -2 2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ9 2 2 -2 2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ10 2 2 0 -2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ53-ζ52 -ζ54+ζ5 -ζ53+ζ52 ζ54-ζ5 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ11 2 2 0 -2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -ζ54+ζ5 -ζ53+ζ52 ζ54-ζ5 ζ53-ζ52 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ12 2 2 0 -2 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ54-ζ5 ζ53-ζ52 -ζ54+ζ5 -ζ53+ζ52 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ13 2 2 0 -2 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -ζ53+ζ52 ζ54-ζ5 ζ53-ζ52 -ζ54+ζ5 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ14 2 -2 0 0 0 2 2 -√-2 √-2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ15 2 -2 0 0 0 2 2 √-2 -√-2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ16 4 -4 0 0 0 -1+√5 -1-√5 0 0 1-√5 1+√5 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ17 4 -4 0 0 0 -1-√5 -1+√5 0 0 1+√5 1-√5 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.D5
On 40 points
Generators in S40
```(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(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)
(1 30 9 25)(2 29 10 24)(3 28 6 23)(4 27 7 22)(5 26 8 21)(11 38 16 33)(12 37 17 32)(13 36 18 31)(14 40 19 35)(15 39 20 34)```

`G:=sub<Sym(40)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30), (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), (1,30,9,25)(2,29,10,24)(3,28,6,23)(4,27,7,22)(5,26,8,21)(11,38,16,33)(12,37,17,32)(13,36,18,31)(14,40,19,35)(15,39,20,34)>;`

`G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30), (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), (1,30,9,25)(2,29,10,24)(3,28,6,23)(4,27,7,22)(5,26,8,21)(11,38,16,33)(12,37,17,32)(13,36,18,31)(14,40,19,35)(15,39,20,34) );`

`G=PermutationGroup([[(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(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)], [(1,30,9,25),(2,29,10,24),(3,28,6,23),(4,27,7,22),(5,26,8,21),(11,38,16,33),(12,37,17,32),(13,36,18,31),(14,40,19,35),(15,39,20,34)]])`

D4.D5 is a maximal subgroup of
D8⋊D5  D83D5  D5×SD16  SD16⋊D5  D4.D10  D4.8D10  D4.9D10  C20.D6  D12.D5  D4.D15  D4.D25  D20.D5  C523SD16  C528SD16
D4.D5 is a maximal quotient of
C20.Q8  C10.Q16  D4⋊Dic5  C20.D6  D12.D5  D4.D15  D4.D25  D20.D5  C523SD16  C528SD16

Matrix representation of D4.D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 37 0 0 21 1
,
 40 0 0 0 18 1 0 0 0 0 40 37 0 0 0 1
,
 16 0 0 0 18 18 0 0 0 0 1 0 0 0 0 1
,
 6 28 0 0 9 35 0 0 0 0 11 22 0 0 28 30
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,21,0,0,37,1],[40,18,0,0,0,1,0,0,0,0,40,0,0,0,37,1],[16,18,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[6,9,0,0,28,35,0,0,0,0,11,28,0,0,22,30] >;`

D4.D5 in GAP, Magma, Sage, TeX

`D_4.D_5`
`% in TeX`

`G:=Group("D4.D5");`
`// GroupNames label`

`G:=SmallGroup(80,16);`
`// by ID`

`G=gap.SmallGroup(80,16);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,40,61,182,97,42,1604]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^5=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽