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## G = C20.12C42order 320 = 26·5

### 5th non-split extension by C20 of C42 acting via C42/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.12C42
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — D10.C23 — C20.12C42
 Lower central C5 — C10 — C20.12C42
 Upper central C1 — C8 — C2×C8

Generators and relations for C20.12C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a3, ac=ca, bc=cb >

Subgroups: 394 in 130 conjugacy classes, 66 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C82M4(2), C8×D5, C2×C52C8, C2×C40, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5, C22⋊F5, C2×C4×D5, C8×F5, C8⋊F5, D5×C2×C8, D5⋊M4(2), D10.C23, C20.12C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, F5, C2×C42, C8○D4, C2×F5, C82M4(2), C4×F5, C22×F5, C2×C4×F5, C20.12C42

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 C4 C8○D4 F5 C2×F5 C2×F5 C4×F5 C4×F5 C20.12C42 kernel C20.12C42 C8×F5 C8⋊F5 D5×C2×C8 D5⋊M4(2) D10.C23 C8×D5 C2×C5⋊2C8 C2×C40 C4.F5 C4⋊F5 C22.F5 C22⋊F5 D5 C2×C8 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 4 4 8 1 2 1 2 2 8

Matrix representation of C20.12C42 in GL4(𝔽41) generated by

 34 7 7 34 7 0 14 14 27 34 27 0 0 27 34 27
,
 9 0 0 0 0 0 0 9 0 9 0 0 32 32 32 32
,
 27 0 0 0 0 27 0 0 0 0 27 0 0 0 0 27
`G:=sub<GL(4,GF(41))| [34,7,27,0,7,0,34,27,7,14,27,34,34,14,0,27],[9,0,0,32,0,0,9,32,0,0,0,32,0,9,0,32],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27] >;`

C20.12C42 in GAP, Magma, Sage, TeX

`C_{20}._{12}C_4^2`
`% in TeX`

`G:=Group("C20.12C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,1056);`
`// by ID`

`G=gap.SmallGroup(320,1056);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,102,6278,1595]);`
`// Polycyclic`

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

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