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## G = Q8.10D10order 160 = 25·5

### 1st non-split extension by Q8 of D10 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8.10D10
 Chief series C1 — C5 — C10 — D10 — C4×D5 — Q8×D5 — Q8.10D10
 Lower central C5 — C10 — Q8.10D10
 Upper central C1 — C2 — C2×Q8

Generators and relations for Q8.10D10
G = < a,b,c,d | a4=1, b2=c10=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c9 >

Subgroups: 392 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2 [×5], C4 [×6], C4 [×4], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], D5 [×4], C10, C10, C2×Q8, C2×Q8 [×4], C4○D4 [×10], Dic5 [×4], C20 [×6], D10 [×4], C2×C10, 2- 1+4, Dic10 [×6], C4×D5 [×12], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8 [×4], C4○D20 [×6], Q8×D5 [×4], Q82D5 [×4], Q8×C10, Q8.10D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2- 1+4, C22×D5 [×7], C23×D5, Q8.10D10

Smallest permutation representation of Q8.10D10
On 80 points
Generators in S80
```(1 43 11 53)(2 44 12 54)(3 45 13 55)(4 46 14 56)(5 47 15 57)(6 48 16 58)(7 49 17 59)(8 50 18 60)(9 51 19 41)(10 52 20 42)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 21 11 31)(2 32 12 22)(3 23 13 33)(4 34 14 24)(5 25 15 35)(6 36 16 26)(7 27 17 37)(8 38 18 28)(9 29 19 39)(10 40 20 30)(41 69 51 79)(42 80 52 70)(43 71 53 61)(44 62 54 72)(45 73 55 63)(46 64 56 74)(47 75 57 65)(48 66 58 76)(49 77 59 67)(50 68 60 78)
(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 80 11 70)(2 69 12 79)(3 78 13 68)(4 67 14 77)(5 76 15 66)(6 65 16 75)(7 74 17 64)(8 63 18 73)(9 72 19 62)(10 61 20 71)(21 42 31 52)(22 51 32 41)(23 60 33 50)(24 49 34 59)(25 58 35 48)(26 47 36 57)(27 56 37 46)(28 45 38 55)(29 54 39 44)(30 43 40 53)```

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

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

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

37 conjugacy classes

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

37 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 D5 D10 D10 2- 1+4 Q8.10D10 kernel Q8.10D10 C4○D20 Q8×D5 Q8⋊2D5 Q8×C10 C2×Q8 C2×C4 Q8 C5 C1 # reps 1 6 4 4 1 2 6 8 1 4

Matrix representation of Q8.10D10 in GL4(𝔽41) generated by

 1 0 7 2 0 1 39 34 17 40 40 0 1 24 0 40
,
 28 8 12 0 33 13 0 12 39 0 13 33 0 39 8 28
,
 3 38 27 27 3 24 14 2 7 7 38 3 34 40 38 17
,
 26 26 36 5 8 15 40 5 6 35 26 26 34 35 8 15
`G:=sub<GL(4,GF(41))| [1,0,17,1,0,1,40,24,7,39,40,0,2,34,0,40],[28,33,39,0,8,13,0,39,12,0,13,8,0,12,33,28],[3,3,7,34,38,24,7,40,27,14,38,38,27,2,3,17],[26,8,6,34,26,15,35,35,36,40,26,8,5,5,26,15] >;`

Q8.10D10 in GAP, Magma, Sage, TeX

`Q_8._{10}D_{10}`
`% in TeX`

`G:=Group("Q8.10D10");`
`// GroupNames label`

`G:=SmallGroup(160,222);`
`// by ID`

`G=gap.SmallGroup(160,222);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,4613]);`
`// Polycyclic`

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

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