Copied to
clipboard

G = D4⋊4D20order 320 = 26·5

3rd semidirect product of D4 and D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4⋊4D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D4⋊8D10 — D4⋊4D20
 Lower central C5 — C10 — C2×C20 — D4⋊4D20
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for D44D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 974 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C52C8, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, D44D4, C40⋊C2, D40, C4.Dic5, D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, Q82D5, C5×C4○D4, D204C4, C20.46D4, C5×C4≀C2, C204D4, C8⋊D10, D4⋊D10, D48D10, D44D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D44D4, C2×D20, D4×D5, C22⋊D20, D44D20

Smallest permutation representation of D44D20
On 40 points
Generators in S40
```(1 19 7 14)(2 20 8 15)(3 16 9 11)(4 17 10 12)(5 18 6 13)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(1 21)(2 37)(3 33)(4 29)(5 25)(6 35)(7 31)(8 27)(9 23)(10 39)(11 28)(12 24)(13 40)(14 36)(15 32)(16 38)(17 34)(18 30)(19 26)(20 22)
(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 5)(2 4)(6 7)(8 10)(11 16)(12 20)(13 19)(14 18)(15 17)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E ··· 20N 20O 20P 40A 40B 40C 40D order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 40 40 40 40 size 1 1 2 4 20 20 20 40 2 2 4 4 4 20 2 2 8 40 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 D20 D20 D4⋊4D4 D4×D5 D4×D5 D4⋊4D20 kernel D4⋊4D20 D20⋊4C4 C20.46D4 C5×C4≀C2 C20⋊4D4 C8⋊D10 D4⋊D10 D4⋊8D10 Dic10 D20 C5×D4 C5×Q8 C22×D5 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of D44D20 in GL4(𝔽41) generated by

 30 9 0 0 32 11 0 0 0 0 11 32 0 0 9 30
,
 0 0 11 32 0 0 9 30 30 9 0 0 32 11 0 0
,
 34 1 0 0 40 0 0 0 0 0 27 30 0 0 11 32
,
 1 34 0 0 0 40 0 0 0 0 11 14 0 0 9 30
`G:=sub<GL(4,GF(41))| [30,32,0,0,9,11,0,0,0,0,11,9,0,0,32,30],[0,0,30,32,0,0,9,11,11,9,0,0,32,30,0,0],[34,40,0,0,1,0,0,0,0,0,27,11,0,0,30,32],[1,0,0,0,34,40,0,0,0,0,11,9,0,0,14,30] >;`

D44D20 in GAP, Magma, Sage, TeX

`D_4\rtimes_4D_{20}`
`% in TeX`

`G:=Group("D4:4D20");`
`// GroupNames label`

`G:=SmallGroup(320,449);`
`// by ID`

`G=gap.SmallGroup(320,449);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,570,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽