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## G = D10.D8order 320 = 26·5

### 2nd non-split extension by D10 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D10.D8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — C40⋊C4 — D10.D8
 Lower central C5 — C10 — C20 — C40 — D10.D8
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D10.D8
G = < a,b,c,d | a10=b2=1, c8=a5, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc7 >

Subgroups: 314 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C4.Q8, M5(2), C4○D8, C52C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, D82C4, C5⋊C16, C8×D5, Dic20, D4.D5, C5×D8, C4⋊F5, D42D5, C8.F5, C40⋊C4, D83D5, D10.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, D82C4, C22⋊F5, D20⋊C4, D10.D8

Character table of D10.D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5 8A 8B 8C 10A 10B 10C 16A 16B 16C 16D 20 40A 40B size 1 1 8 10 2 10 40 40 40 4 4 10 10 4 16 16 20 20 20 20 8 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 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 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 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 linear of order 2 ρ5 1 1 1 -1 1 -1 -i -1 i 1 1 -1 -1 1 1 1 -i i i -i 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -i 1 i 1 1 -1 -1 1 -1 -1 i -i -i i 1 1 1 linear of order 4 ρ7 1 1 1 -1 1 -1 i -1 -i 1 1 -1 -1 1 1 1 i -i -i i 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 i 1 -i 1 1 -1 -1 1 -1 -1 -i i i -i 1 1 1 linear of order 4 ρ9 2 2 0 2 2 2 0 0 0 2 -2 -2 -2 2 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 0 -2 2 -2 0 0 0 2 -2 2 2 2 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ11 2 2 0 2 -2 -2 0 0 0 2 0 0 0 2 0 0 √2 -√2 √2 -√2 -2 0 0 orthogonal lifted from D8 ρ12 2 2 0 2 -2 -2 0 0 0 2 0 0 0 2 0 0 -√2 √2 -√2 √2 -2 0 0 orthogonal lifted from D8 ρ13 2 2 0 -2 -2 2 0 0 0 2 0 0 0 2 0 0 -√-2 -√-2 √-2 √-2 -2 0 0 complex lifted from SD16 ρ14 2 2 0 -2 -2 2 0 0 0 2 0 0 0 2 0 0 √-2 √-2 -√-2 -√-2 -2 0 0 complex lifted from SD16 ρ15 4 4 -4 0 4 0 0 0 0 -1 4 0 0 -1 1 1 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×F5 ρ16 4 4 4 0 4 0 0 0 0 -1 4 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 0 4 0 0 0 0 -1 -4 0 0 -1 -√5 √5 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ18 4 4 0 0 4 0 0 0 0 -1 -4 0 0 -1 √5 -√5 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ19 4 -4 0 0 0 0 0 0 0 4 0 -2√-2 2√-2 -4 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4 ρ20 4 -4 0 0 0 0 0 0 0 4 0 2√-2 -2√-2 -4 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4 ρ21 8 8 0 0 -8 0 0 0 0 -2 0 0 0 -2 0 0 0 0 0 0 2 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ22 8 -8 0 0 0 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 -√10 √10 symplectic faithful, Schur index 2 ρ23 8 -8 0 0 0 0 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 √10 -√10 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D10.D8 in GL8(𝔽241)

 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 0 0 0 0 1 0 240 0 0 0 0 0 0 1 240 0
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 42 177 0 1 0 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 1 0 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 240 1
,
 166 11 0 203 0 0 0 0 166 11 38 203 0 0 0 0 1 0 0 0 0 0 0 0 19 1 11 64 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 240 0 0
,
 1 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 166 11 0 203 0 0 0 0 220 209 222 0 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 1 0 0

`G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0],[240,0,0,42,0,0,0,0,0,240,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1],[166,166,1,19,0,0,0,0,11,11,0,1,0,0,0,0,0,38,0,11,0,0,0,0,203,203,0,64,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0],[1,0,166,220,0,0,0,0,0,240,11,209,0,0,0,0,0,0,0,222,0,0,0,0,0,0,203,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;`

D10.D8 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,241);`
`// by ID`

`G=gap.SmallGroup(320,241);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,675,794,80,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=1,c^8=a^5,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^7>;`
`// generators/relations`

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