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## G = C20.53D4order 160 = 25·5

### 10th non-split extension by C20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.53D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C20.53D4
 Lower central C5 — C10 — C20 — C20.53D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C20.53D4
G = < a,b,c | a20=1, b4=a10, c2=a5, bab-1=cac-1=a9, cbc-1=a10b3 >

Smallest permutation representation of C20.53D4
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 52 16 47 11 42 6 57)(2 41 17 56 12 51 7 46)(3 50 18 45 13 60 8 55)(4 59 19 54 14 49 9 44)(5 48 20 43 15 58 10 53)(21 79 26 64 31 69 36 74)(22 68 27 73 32 78 37 63)(23 77 28 62 33 67 38 72)(24 66 29 71 34 76 39 61)(25 75 30 80 35 65 40 70)
(1 69 6 74 11 79 16 64)(2 78 7 63 12 68 17 73)(3 67 8 72 13 77 18 62)(4 76 9 61 14 66 19 71)(5 65 10 70 15 75 20 80)(21 47 26 52 31 57 36 42)(22 56 27 41 32 46 37 51)(23 45 28 50 33 55 38 60)(24 54 29 59 34 44 39 49)(25 43 30 48 35 53 40 58)```

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

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

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

C20.53D4 is a maximal subgroup of
D20.2D4  D20.3D4  D20.6D4  D20.7D4  M4(2).22D10  C42.196D10  D5×C8.C4  M4(2).25D10  C23.Dic10  C40.93D4  C40.50D4  M4(2).D10  M4(2).13D10  M4(2).15D10  M4(2).16D10  C60.105D4  C60.D4  C60.210D4
C20.53D4 is a maximal quotient of
C20.53D8  C20.39SD16  C20.34C42  C60.105D4  C60.D4  C60.210D4

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C4 D4 Q8 D5 D10 C8.C4 C4×D5 C5⋊D4 Dic10 C20.53D4 kernel C20.53D4 C2×C5⋊2C8 C4.Dic5 C5×M4(2) C5⋊2C8 C20 C2×C10 M4(2) C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 4 4 4 4 4

Matrix representation of C20.53D4 in GL4(𝔽41) generated by

 34 1 0 0 33 1 0 0 0 0 9 0 0 0 0 9
,
 12 27 0 0 25 29 0 0 0 0 27 0 0 0 31 38
,
 12 2 0 0 30 29 0 0 0 0 4 12 0 0 37 37
`G:=sub<GL(4,GF(41))| [34,33,0,0,1,1,0,0,0,0,9,0,0,0,0,9],[12,25,0,0,27,29,0,0,0,0,27,31,0,0,0,38],[12,30,0,0,2,29,0,0,0,0,4,37,0,0,12,37] >;`

C20.53D4 in GAP, Magma, Sage, TeX

`C_{20}._{53}D_4`
`% in TeX`

`G:=Group("C20.53D4");`
`// GroupNames label`

`G:=SmallGroup(160,29);`
`// by ID`

`G=gap.SmallGroup(160,29);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,121,31,86,297,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=a^10,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^3>;`
`// generators/relations`

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