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## G = D4.10D20order 320 = 26·5

### 5th non-split extension by D4 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.10D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.10D10 — D4.10D20
 Lower central C5 — C10 — C2×C20 — D4.10D20
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for D4.10D20
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a5c-1 >

Subgroups: 686 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C52C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4.8D4, C40⋊C2, D40, C4.Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×Dic10, C2×Dic10, C2×D20, C4○D20, C4○D20, D42D5, Q8×D5, C5×C4○D4, D204C4, C4.12D20, C5×C4≀C2, C4.D20, C8⋊D10, D4⋊D10, D4.10D10, D4.10D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.8D4, C2×D20, D4×D5, C22⋊D20, D4.10D20

Smallest permutation representation of D4.10D20
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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 80)(57 79)(58 78)(59 77)(60 76)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 50 31 60)(22 51 32 41)(23 52 33 42)(24 53 34 43)(25 54 35 44)(26 55 36 45)(27 56 37 46)(28 57 38 47)(29 58 39 48)(30 59 40 49)
(1 60 11 50)(2 41 12 51)(3 42 13 52)(4 43 14 53)(5 44 15 54)(6 45 16 55)(7 46 17 56)(8 47 18 57)(9 48 19 58)(10 49 20 59)(21 62 31 72)(22 63 32 73)(23 64 33 74)(24 65 34 75)(25 66 35 76)(26 67 36 77)(27 68 37 78)(28 69 38 79)(29 70 39 80)(30 71 40 61)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 D20 D20 D4.8D4 D4×D5 D4×D5 D4.10D20 kernel D4.10D20 D20⋊4C4 C4.12D20 C5×C4≀C2 C4.D20 C8⋊D10 D4⋊D10 D4.10D10 Dic10 D20 C2×Dic5 C5×D4 C5×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 4 4 2 2 2 8

Matrix representation of D4.10D20 in GL4(𝔽41) generated by

 2 14 0 0 11 16 0 0 0 0 2 14 0 0 11 16
,
 0 0 39 4 0 0 30 2 39 4 0 0 30 2 0 0
,
 18 5 0 0 1 23 0 0 0 0 32 0 0 0 0 32
,
 0 0 23 36 0 0 40 18 18 5 0 0 1 23 0 0
`G:=sub<GL(4,GF(41))| [2,11,0,0,14,16,0,0,0,0,2,11,0,0,14,16],[0,0,39,30,0,0,4,2,39,30,0,0,4,2,0,0],[18,1,0,0,5,23,0,0,0,0,32,0,0,0,0,32],[0,0,18,1,0,0,5,23,23,40,0,0,36,18,0,0] >;`

D4.10D20 in GAP, Magma, Sage, TeX

`D_4._{10}D_{20}`
`% in TeX`

`G:=Group("D4.10D20");`
`// GroupNames label`

`G:=SmallGroup(320,454);`
`// by ID`

`G=gap.SmallGroup(320,454);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,136,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;`
`// generators/relations`

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