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## G = C20.25C42order 320 = 26·5

### 18th non-split extension by C20 of C42 acting via C42/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.25C42
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C2×C4×D5 — D5⋊M4(2) — C20.25C42
 Lower central C5 — C20 — C20.25C42
 Upper central C1 — C4 — C2×C8

Generators and relations for C20.25C42
G = < a,b,c | a20=1, b4=c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

Subgroups: 322 in 86 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C4.10C42, C8⋊D5, C2×C52C8, C2×C40, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×C8⋊D5, D5⋊M4(2), C20.25C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, C4.10C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C20.25C42

Smallest permutation representation of C20.25C42
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 48 16 43 11 58 6 53)(2 45 5 56 12 55 15 46)(3 42 14 49 13 52 4 59)(7 50 10 41 17 60 20 51)(8 47 19 54 18 57 9 44)(21 71 26 76 31 61 36 66)(22 68 35 69 32 78 25 79)(23 65 24 62 33 75 34 72)(27 73 40 74 37 63 30 64)(28 70 29 67 38 80 39 77)
(1 21 16 36 11 31 6 26)(2 22 17 37 12 32 7 27)(3 23 18 38 13 33 8 28)(4 24 19 39 14 34 9 29)(5 25 20 40 15 35 10 30)(41 79 46 64 51 69 56 74)(42 80 47 65 52 70 57 75)(43 61 48 66 53 71 58 76)(44 62 49 67 54 72 59 77)(45 63 50 68 55 73 60 78)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C4 C4 C4 D4 D4 Q8 F5 C2×F5 C4.10C42 C4×F5 C22⋊F5 C4⋊F5 C20.25C42 kernel C20.25C42 C2×C8⋊D5 D5⋊M4(2) C2×C5⋊2C8 C2×C40 D5⋊C8 C4×D5 C2×Dic5 C22×D5 C2×C8 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 2 2 2 8 2 1 1 1 1 2 2 2 2 8

Matrix representation of C20.25C42 in GL4(𝔽41) generated by

 13 32 0 0 9 0 0 0 0 26 28 28 26 40 13 32
,
 38 34 39 0 14 31 0 39 1 25 3 7 18 17 27 10
,
 6 39 0 0 2 35 0 0 27 0 13 23 27 14 18 28
`G:=sub<GL(4,GF(41))| [13,9,0,26,32,0,26,40,0,0,28,13,0,0,28,32],[38,14,1,18,34,31,25,17,39,0,3,27,0,39,7,10],[6,2,27,27,39,35,0,14,0,0,13,18,0,0,23,28] >;`

C20.25C42 in GAP, Magma, Sage, TeX

`C_{20}._{25}C_4^2`
`% in TeX`

`G:=Group("C20.25C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,235);`
`// by ID`

`G=gap.SmallGroup(320,235);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1123,136,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^5*b>;`
`// generators/relations`

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