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## G = D20.2C4order 160 = 25·5

### The non-split extension by D20 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.2C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D20.2C4
 Lower central C5 — C10 — D20.2C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D20.2C4
G = < a,b,c | a20=b2=1, c4=a10, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 160 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, M4(2), M4(2), C4○D4, Dic5, C20, D10, C2×C10, C8○D4, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C8×D5, C8⋊D5, C2×C52C8, C5×M4(2), C4○D20, D20.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, D10, C8○D4, C4×D5, C22×D5, C2×C4×D5, D20.2C4

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

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

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

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

40 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 10B 10C 10D 20A 20B 20C 20D 20E 20F 40A ··· 40H 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 10 10 20 20 20 20 20 20 40 ··· 40 size 1 1 2 10 10 1 1 2 10 10 2 2 2 2 2 2 5 5 5 5 10 10 2 2 4 4 2 2 2 2 4 4 4 ··· 4

40 irreducible representations

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

Matrix representation of D20.2C4 in GL4(𝔽41) generated by

 0 40 0 0 1 35 0 0 0 0 27 34 0 0 34 14
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 4 1
,
 1 0 0 0 0 1 0 0 0 0 1 21 0 0 16 40
`G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,27,34,0,0,34,14],[0,1,0,0,1,0,0,0,0,0,40,4,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,21,40] >;`

D20.2C4 in GAP, Magma, Sage, TeX

`D_{20}._2C_4`
`% in TeX`

`G:=Group("D20.2C4");`
`// GroupNames label`

`G:=SmallGroup(160,128);`
`// by ID`

`G=gap.SmallGroup(160,128);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,188,50,69,4613]);`
`// Polycyclic`

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

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