Copied to
clipboard

## G = Q16⋊F5order 320 = 26·5

### 4th semidirect product of Q16 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q16⋊F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — Q8.F5 — Q16⋊F5
 Lower central C5 — C10 — C20 — Q16⋊F5
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊F5
G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a6b, dcd-1=c3 >

Subgroups: 418 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2 [×3], C4, C4 [×4], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4 [×4], Q8 [×2], D5 [×3], C10, C42, C2×C8 [×4], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], F5, D10, D10 [×2], C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], C4×D5, C4×D5 [×2], D20 [×2], D20 [×2], C5×Q8 [×2], C2×F5, C8.26D4, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, D5⋊C8, D5⋊C8 [×2], C4.F5 [×2], C4.F5 [×2], C4×F5, Q82D5 [×2], C8⋊F5, C40.C4, Q82F5 [×2], Q8.D10, Q8.F5 [×2], Q16⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8.26D4, C22×F5, D4×F5, Q16⋊F5

Character table of Q16⋊F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10 20A 20B 20C 40A 40B size 1 1 10 20 20 2 4 4 5 5 20 20 4 4 10 10 10 10 20 20 20 20 20 4 8 16 16 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 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 -1 -1 -1 1 1 1 -1 -1 -i i 1 1 -i i -i i -1 i -i i -i 1 1 1 1 1 1 linear of order 4 ρ10 1 1 -1 -1 1 1 -1 1 -1 -1 -i i 1 -1 i -i i -i 1 i -i -i i 1 1 -1 1 -1 -1 linear of order 4 ρ11 1 1 -1 1 -1 1 1 -1 -1 -1 -i i 1 -1 i -i i -i 1 -i i i -i 1 1 1 -1 -1 -1 linear of order 4 ρ12 1 1 -1 1 1 1 -1 -1 -1 -1 -i i 1 1 -i i -i i -1 -i i -i i 1 1 -1 -1 1 1 linear of order 4 ρ13 1 1 -1 -1 1 1 -1 1 -1 -1 i -i 1 -1 -i i -i i 1 -i i i -i 1 1 -1 1 -1 -1 linear of order 4 ρ14 1 1 -1 -1 -1 1 1 1 -1 -1 i -i 1 1 i -i i -i -1 -i i -i i 1 1 1 1 1 1 linear of order 4 ρ15 1 1 -1 1 1 1 -1 -1 -1 -1 i -i 1 1 i -i i -i -1 i -i i -i 1 1 -1 -1 1 1 linear of order 4 ρ16 1 1 -1 1 -1 1 1 -1 -1 -1 i -i 1 -1 -i i -i i 1 i -i -i i 1 1 1 -1 -1 -1 linear of order 4 ρ17 2 2 -2 0 0 -2 0 0 2 2 0 0 2 0 2 2 -2 -2 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 0 -2 0 0 2 2 0 0 2 0 -2 -2 2 2 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 0 0 -2 0 0 -2 -2 0 0 2 0 -2i 2i 2i -2i 0 0 0 0 0 2 -2 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 0 0 -2 0 0 -2 -2 0 0 2 0 2i -2i -2i 2i 0 0 0 0 0 2 -2 0 0 0 0 complex lifted from C4○D4 ρ21 4 4 0 0 0 4 4 -4 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 0 0 0 4 -4 -4 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ23 4 4 0 0 0 4 4 4 0 0 0 0 -1 4 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 0 0 0 4 -4 4 0 0 0 0 -1 -4 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from C2×F5 ρ25 4 -4 0 0 0 0 0 0 4i -4i 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ26 4 -4 0 0 0 0 0 0 -4i 4i 0 0 4 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 complex lifted from C8.26D4 ρ27 8 8 0 0 0 -8 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4×F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 -√10 √10 orthogonal faithful ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 √10 -√10 orthogonal faithful

Smallest permutation representation of Q16⋊F5
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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 75 41 12 66)(2 76 42 13 67)(3 77 43 14 68)(4 78 44 15 69)(5 79 45 16 70)(6 80 46 9 71)(7 73 47 10 72)(8 74 48 11 65)(17 40 50 60 29)(18 33 51 61 30)(19 34 52 62 31)(20 35 53 63 32)(21 36 54 64 25)(22 37 55 57 26)(23 38 56 58 27)(24 39 49 59 28)
(2 6)(4 8)(9 76 46 67)(10 73 47 72)(11 78 48 69)(12 75 41 66)(13 80 42 71)(14 77 43 68)(15 74 44 65)(16 79 45 70)(17 56 36 31)(18 53 37 28)(19 50 38 25)(20 55 39 30)(21 52 40 27)(22 49 33 32)(23 54 34 29)(24 51 35 26)(57 59 61 63)(58 64 62 60)```

`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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,40,50,60,29)(18,33,51,61,30)(19,34,52,62,31)(20,35,53,63,32)(21,36,54,64,25)(22,37,55,57,26)(23,38,56,58,27)(24,39,49,59,28), (2,6)(4,8)(9,76,46,67)(10,73,47,72)(11,78,48,69)(12,75,41,66)(13,80,42,71)(14,77,43,68)(15,74,44,65)(16,79,45,70)(17,56,36,31)(18,53,37,28)(19,50,38,25)(20,55,39,30)(21,52,40,27)(22,49,33,32)(23,54,34,29)(24,51,35,26)(57,59,61,63)(58,64,62,60)>;`

`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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(17,45,21,41)(18,44,22,48)(19,43,23,47)(20,42,24,46)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,75,41,12,66)(2,76,42,13,67)(3,77,43,14,68)(4,78,44,15,69)(5,79,45,16,70)(6,80,46,9,71)(7,73,47,10,72)(8,74,48,11,65)(17,40,50,60,29)(18,33,51,61,30)(19,34,52,62,31)(20,35,53,63,32)(21,36,54,64,25)(22,37,55,57,26)(23,38,56,58,27)(24,39,49,59,28), (2,6)(4,8)(9,76,46,67)(10,73,47,72)(11,78,48,69)(12,75,41,66)(13,80,42,71)(14,77,43,68)(15,74,44,65)(16,79,45,70)(17,56,36,31)(18,53,37,28)(19,50,38,25)(20,55,39,30)(21,52,40,27)(22,49,33,32)(23,54,34,29)(24,51,35,26)(57,59,61,63)(58,64,62,60) );`

`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)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(17,45,21,41),(18,44,22,48),(19,43,23,47),(20,42,24,46),(25,75,29,79),(26,74,30,78),(27,73,31,77),(28,80,32,76),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)], [(1,75,41,12,66),(2,76,42,13,67),(3,77,43,14,68),(4,78,44,15,69),(5,79,45,16,70),(6,80,46,9,71),(7,73,47,10,72),(8,74,48,11,65),(17,40,50,60,29),(18,33,51,61,30),(19,34,52,62,31),(20,35,53,63,32),(21,36,54,64,25),(22,37,55,57,26),(23,38,56,58,27),(24,39,49,59,28)], [(2,6),(4,8),(9,76,46,67),(10,73,47,72),(11,78,48,69),(12,75,41,66),(13,80,42,71),(14,77,43,68),(15,74,44,65),(16,79,45,70),(17,56,36,31),(18,53,37,28),(19,50,38,25),(20,55,39,30),(21,52,40,27),(22,49,33,32),(23,54,34,29),(24,51,35,26),(57,59,61,63),(58,64,62,60)])`

Matrix representation of Q16⋊F5 in GL8(𝔽41)

 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 0 0 0 0 0 0 0 0 0 0 0 33 0 0 0 0 0 20 31 0 0 0 0 0 0 4 21 0 0 0 0 0 4 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 40 0 0 37 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0
,
 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 33 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 40

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,20,4,0,0,0,0,0,0,31,21,0,0,0,0,0,33,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,32,0,0,0,0,0,37,32,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,33,32,0,0,0,0,0,0,0,0,40] >;`

Q16⋊F5 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes F_5`
`% in TeX`

`G:=Group("Q16:F5");`
`// GroupNames label`

`G:=SmallGroup(320,1079);`
`// by ID`

`G=gap.SmallGroup(320,1079);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,851,438,102,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^5=d^4=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;`
`// generators/relations`

Export

׿
×
𝔽