Copied to
clipboard

## G = D8.D10order 320 = 26·5

### 1st non-split extension by D8 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8.D10
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — D40⋊7C2 — D8.D10
 Lower central C5 — C10 — C20 — C40 — D8.D10
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×D8

Generators and relations for D8.D10
G = < a,b,c,d | a8=b2=1, c10=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c9 >

Subgroups: 398 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, M5(2), D16, SD32, C2×D8, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C22×C10, C16⋊C22, C52C16, C40⋊C2, D40, Dic20, C2×C40, C5×D8, C5×D8, C4○D20, D4×C10, C20.4C8, C5⋊D16, D8.D5, D407C2, C10×D8, D8.D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, C16⋊C22, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, D8.D10

Smallest permutation representation of D8.D10
On 80 points
Generators in S80
```(1 38 6 23 11 28 16 33)(2 39 7 24 12 29 17 34)(3 40 8 25 13 30 18 35)(4 21 9 26 14 31 19 36)(5 22 10 27 15 32 20 37)(41 63 56 78 51 73 46 68)(42 64 57 79 52 74 47 69)(43 65 58 80 53 75 48 70)(44 66 59 61 54 76 49 71)(45 67 60 62 55 77 50 72)
(1 33)(2 24)(3 35)(4 26)(5 37)(6 28)(7 39)(8 30)(9 21)(10 32)(11 23)(12 34)(13 25)(14 36)(15 27)(16 38)(17 29)(18 40)(19 31)(20 22)(41 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)(62 72)(64 74)(66 76)(68 78)(70 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 57 11 47)(2 46 12 56)(3 55 13 45)(4 44 14 54)(5 53 15 43)(6 42 16 52)(7 51 17 41)(8 60 18 50)(9 49 19 59)(10 58 20 48)(21 71 31 61)(22 80 32 70)(23 69 33 79)(24 78 34 68)(25 67 35 77)(26 76 36 66)(27 65 37 75)(28 74 38 64)(29 63 39 73)(30 72 40 62)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 8C 10A ··· 10F 10G ··· 10N 16A 16B 16C 16D 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 5 5 8 8 8 10 ··· 10 10 ··· 10 16 16 16 16 20 20 20 20 40 ··· 40 size 1 1 2 8 8 40 2 2 40 2 2 2 2 4 2 ··· 2 8 ··· 8 20 20 20 20 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 C5⋊D4 C5⋊D4 C16⋊C22 D4⋊D5 D4⋊D5 D8.D10 kernel D8.D10 C20.4C8 C5⋊D16 D8.D5 D40⋊7C2 C10×D8 C40 C2×C20 C2×D8 C20 C2×C10 C2×C8 D8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of D8.D10 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 219 219 0 0 0 0 11 0 0 0 0 0 0 0 0 22 0 0 0 0 230 219
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 22 0 0 0 0 11 0 0 0 0 0 0 0 240 0 0 0 0 0 1 1
,
 36 0 0 0 0 0 92 154 0 0 0 0 0 0 1 2 0 0 0 0 240 240 0 0 0 0 0 0 1 2 0 0 0 0 240 240
,
 187 46 0 0 0 0 120 54 0 0 0 0 0 0 0 0 1 2 0 0 0 0 240 240 0 0 1 2 0 0 0 0 240 240 0 0

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,219,11,0,0,0,0,219,0,0,0,0,0,0,0,0,230,0,0,0,0,22,219],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,11,0,0,0,0,22,0,0,0,0,0,0,0,240,1,0,0,0,0,0,1],[36,92,0,0,0,0,0,154,0,0,0,0,0,0,1,240,0,0,0,0,2,240,0,0,0,0,0,0,1,240,0,0,0,0,2,240],[187,120,0,0,0,0,46,54,0,0,0,0,0,0,0,0,1,240,0,0,0,0,2,240,0,0,1,240,0,0,0,0,2,240,0,0] >;`

D8.D10 in GAP, Magma, Sage, TeX

`D_8.D_{10}`
`% in TeX`

`G:=Group("D8.D10");`
`// GroupNames label`

`G:=SmallGroup(320,774);`
`// by ID`

`G=gap.SmallGroup(320,774);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,675,185,192,1684,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^10=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
×
𝔽