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## G = C20.24C42order 320 = 26·5

### 17th non-split extension by C20 of C42 acting via C42/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.24C42
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C2×C4×D5 — D10.C23 — C20.24C42
 Lower central C5 — C20 — C20.24C42
 Upper central C1 — C4 — C2×C8

Generators and relations for C20.24C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

Subgroups: 418 in 94 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C42⋊C2, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4.9C42, C8⋊D5, C2×C52C8, C2×C40, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×C8⋊D5, D10.C23, C20.24C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, C4.9C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C20.24C42

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F ··· 4M 5 8A 8B 8C 8D 10A 10B 10C 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 4 4 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 20 ··· 20 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 Q8 D4 F5 C2×F5 C4.9C42 C4×F5 C22⋊F5 C4⋊F5 C20.24C42 kernel C20.24C42 C2×C8⋊D5 D10.C23 C2×C5⋊2C8 C2×C40 C4×F5 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 C20.24C42 in GL4(𝔽41) generated by

 32 32 32 32 9 0 0 0 0 9 0 0 0 0 9 0
,
 34 0 27 27 27 27 0 34 14 7 14 0 7 34 34 7
,
 30 19 6 13 28 17 6 34 7 35 24 13 28 35 22 11
`G:=sub<GL(4,GF(41))| [32,9,0,0,32,0,9,0,32,0,0,9,32,0,0,0],[34,27,14,7,0,27,7,34,27,0,14,34,27,34,0,7],[30,28,7,28,19,17,35,35,6,6,24,22,13,34,13,11] >;`

C20.24C42 in GAP, Magma, Sage, TeX

`C_{20}._{24}C_4^2`
`% in TeX`

`G:=Group("C20.24C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,233);`
`// by ID`

`G=gap.SmallGroup(320,233);`
`# by ID`

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

`G:=Group<a,b,c|a^20=b^4=1,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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