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## G = D4.10D10order 160 = 25·5

### The non-split extension by D4 of D10 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4.10D10
 Chief series C1 — C5 — C10 — D10 — C4×D5 — Q8×D5 — D4.10D10
 Lower central C5 — C10 — D4.10D10
 Upper central C1 — C2 — C4○D4

Generators and relations for D4.10D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c9 >

Subgroups: 376 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×Dic10, C4○D20, D42D5, Q8×D5, C5×C4○D4, D4.10D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C22×D5, C23×D5, D4.10D10

Smallest permutation representation of D4.10D10
On 80 points
Generators in S80
```(1 68 11 78)(2 79 12 69)(3 70 13 80)(4 61 14 71)(5 72 15 62)(6 63 16 73)(7 74 17 64)(8 65 18 75)(9 76 19 66)(10 67 20 77)(21 51 31 41)(22 42 32 52)(23 53 33 43)(24 44 34 54)(25 55 35 45)(26 46 36 56)(27 57 37 47)(28 48 38 58)(29 59 39 49)(30 50 40 60)
(1 23)(2 34)(3 25)(4 36)(5 27)(6 38)(7 29)(8 40)(9 31)(10 22)(11 33)(12 24)(13 35)(14 26)(15 37)(16 28)(17 39)(18 30)(19 21)(20 32)(41 66)(42 77)(43 68)(44 79)(45 70)(46 61)(47 72)(48 63)(49 74)(50 65)(51 76)(52 67)(53 78)(54 69)(55 80)(56 71)(57 62)(58 73)(59 64)(60 75)
(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 47 11 57)(2 56 12 46)(3 45 13 55)(4 54 14 44)(5 43 15 53)(6 52 16 42)(7 41 17 51)(8 50 18 60)(9 59 19 49)(10 48 20 58)(21 74 31 64)(22 63 32 73)(23 72 33 62)(24 61 34 71)(25 70 35 80)(26 79 36 69)(27 68 37 78)(28 77 38 67)(29 66 39 76)(30 75 40 65)```

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

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

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

37 conjugacy classes

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

37 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D5 D10 D10 D10 2- 1+4 D4.10D10 kernel D4.10D10 C2×Dic10 C4○D20 D4⋊2D5 Q8×D5 C5×C4○D4 C4○D4 C2×C4 D4 Q8 C5 C1 # reps 1 3 3 6 2 1 2 6 6 2 1 4

Matrix representation of D4.10D10 in GL4(𝔽41) generated by

 2 0 6 0 0 2 0 6 6 0 39 0 0 6 0 39
,
 25 13 19 23 28 16 18 22 19 23 16 28 18 22 13 25
,
 0 0 7 7 0 0 34 40 34 34 0 0 7 1 0 0
,
 0 0 14 27 0 0 11 27 27 14 0 0 30 14 0 0
`G:=sub<GL(4,GF(41))| [2,0,6,0,0,2,0,6,6,0,39,0,0,6,0,39],[25,28,19,18,13,16,23,22,19,18,16,13,23,22,28,25],[0,0,34,7,0,0,34,1,7,34,0,0,7,40,0,0],[0,0,27,30,0,0,14,14,14,11,0,0,27,27,0,0] >;`

D4.10D10 in GAP, Magma, Sage, TeX

`D_4._{10}D_{10}`
`% in TeX`

`G:=Group("D4.10D10");`
`// GroupNames label`

`G:=SmallGroup(160,225);`
`// by ID`

`G=gap.SmallGroup(160,225);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,69,4613]);`
`// Polycyclic`

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

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