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## G = D20.3C4order 160 = 25·5

### The non-split extension by D20 of C4 acting through Inn(D20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.3C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D20.3C4
 Lower central C5 — C10 — D20.3C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D20.3C4
G = < a,b,c | a20=b2=1, c4=a10, bab=a-1, ac=ca, bc=cb >

Subgroups: 160 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, D20.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, D10 [×3], C8○D4, C4×D5 [×2], C22×D5, C2×C4×D5, D20.3C4

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

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

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

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

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D5 D10 D10 C8○D4 C4×D5 C4×D5 D20.3C4 kernel D20.3C4 C8×D5 C8⋊D5 C4.Dic5 C2×C40 C4○D20 Dic10 D20 C5⋊D4 C2×C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 4 2 4 2 4 4 4 16

Matrix representation of D20.3C4 in GL2(𝔽41) generated by

 14 39 16 30
,
 1 1 0 40
,
 14 0 0 14
`G:=sub<GL(2,GF(41))| [14,16,39,30],[1,0,1,40],[14,0,0,14] >;`

D20.3C4 in GAP, Magma, Sage, TeX

`D_{20}._3C_4`
`% in TeX`

`G:=Group("D20.3C4");`
`// GroupNames label`

`G:=SmallGroup(160,122);`
`// by ID`

`G=gap.SmallGroup(160,122);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,50,69,4613]);`
`// Polycyclic`

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

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