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## G = C4⋊D20order 160 = 25·5

### The semidirect product of C4 and D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊D20
G = < a,b,c | a4=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 384 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C4⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C22×D5, C2×D20, D4×D5, Q82D5, C4⋊D20

Smallest permutation representation of C4⋊D20
On 80 points
Generators in S80
```(1 71 41 26)(2 27 42 72)(3 73 43 28)(4 29 44 74)(5 75 45 30)(6 31 46 76)(7 77 47 32)(8 33 48 78)(9 79 49 34)(10 35 50 80)(11 61 51 36)(12 37 52 62)(13 63 53 38)(14 39 54 64)(15 65 55 40)(16 21 56 66)(17 67 57 22)(18 23 58 68)(19 69 59 24)(20 25 60 70)
(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)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D20 D4×D5 Q8⋊2D5 kernel C4⋊D20 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C2×D20 C20 D10 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 2 2 2 2 6 8 2 2

Matrix representation of C4⋊D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 39 0 0 1 40
,
 14 39 0 0 16 30 0 0 0 0 1 0 0 0 1 40
,
 1 1 0 0 0 40 0 0 0 0 1 0 0 0 1 40
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[14,16,0,0,39,30,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,1,1,0,0,0,40] >;`

C4⋊D20 in GAP, Magma, Sage, TeX

`C_4\rtimes D_{20}`
`% in TeX`

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

`G:=SmallGroup(160,116);`
`// by ID`

`G=gap.SmallGroup(160,116);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,50,4613]);`
`// Polycyclic`

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

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