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## G = D20⋊8C4order 160 = 25·5

### 5th semidirect product of D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20⋊8C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D20⋊8C4
 Lower central C5 — C10 — D20⋊8C4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D208C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 312 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, D208C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C22×D5, C2×C4×D5, D4×D5, Q82D5, D208C4

Smallest permutation representation of D208C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)
(1 77 55 29)(2 68 56 40)(3 79 57 31)(4 70 58 22)(5 61 59 33)(6 72 60 24)(7 63 41 35)(8 74 42 26)(9 65 43 37)(10 76 44 28)(11 67 45 39)(12 78 46 30)(13 69 47 21)(14 80 48 32)(15 71 49 23)(16 62 50 34)(17 73 51 25)(18 64 52 36)(19 75 53 27)(20 66 54 38)```

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 C4×D5 D4×D5 Q8⋊2D5 kernel D20⋊8C4 C4×Dic5 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C2×D20 D20 Dic5 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 1 2 1 2 1 8 2 2 2 6 8 2 2

Matrix representation of D208C4 in GL5(𝔽41)

 40 0 0 0 0 0 6 40 0 0 0 36 1 0 0 0 0 0 40 21 0 0 0 37 1
,
 40 0 0 0 0 0 40 40 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 37 1
,
 32 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 9 16 0 0 0 0 32

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,36,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,21,1],[40,0,0,0,0,0,40,0,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,0,1],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,16,32] >;`

D208C4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_8C_4`
`% in TeX`

`G:=Group("D20:8C4");`
`// GroupNames label`

`G:=SmallGroup(160,114);`
`// by ID`

`G=gap.SmallGroup(160,114);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,188,50,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;`
`// generators/relations`

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