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## G = D10.10D8order 320 = 26·5

### 6th non-split extension by D10 of D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D10.10D8
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C2×C4×D5 — C2×C4⋊F5 — D10.10D8
 Lower central C5 — C10 — C20 — D10.10D8
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for D10.10D8
G = < a,b,c,d | a10=b2=c8=1, d2=a4b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=a5c-1 >

Subgroups: 514 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C4⋊C4, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22.4Q16, C8×D5, C2×C52C8, C2×C40, C4⋊F5, C4⋊F5, C2×C4×D5, C22×F5, D5×C2×C8, C2×C4⋊F5, D10.10D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, F5, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×F5, C22.4Q16, C4×F5, C4⋊F5, C22⋊F5, C40⋊C4, D5.D8, D10.3Q8, D10.10D8

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + - + + - + + + image C1 C2 C2 C4 C4 C4 D4 Q8 D4 D8 SD16 Q16 F5 C2×F5 C4×F5 C22⋊F5 C4⋊F5 C40⋊C4 D5.D8 kernel D10.10D8 D5×C2×C8 C2×C4⋊F5 C2×C5⋊2C8 C2×C40 C4⋊F5 C4×D5 C2×Dic5 C22×D5 D10 D10 D10 C2×C8 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 2 2 2 8 2 1 1 2 4 2 1 1 2 2 2 4 4

Matrix representation of D10.10D8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 1 1 1 0 0 40 0 0 0 0 0 0 40 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40 0 0 0
,
 22 22 0 0 0 0 15 2 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 34 34 0 0 0 0 13 7 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 32 0 0 0 0 9 9 9 9

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[22,15,0,0,0,0,22,2,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[34,13,0,0,0,0,34,7,0,0,0,0,0,0,32,0,0,9,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,32,0,9] >;`

D10.10D8 in GAP, Magma, Sage, TeX

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

`G:=Group("D10.10D8");`
`// GroupNames label`

`G:=SmallGroup(320,231);`
`// by ID`

`G=gap.SmallGroup(320,231);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1123,136,6278,3156]);`
`// Polycyclic`

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

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