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## G = C20.33C42order 320 = 26·5

### 3rd non-split extension by C20 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.33C42
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C23.21D10 — C20.33C42
 Lower central C5 — C10 — C20 — C20.33C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C20.33C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, cac-1=a11, cbc-1=a5b >

Subgroups: 358 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C40, C2×Dic5, C2×C20, C22×C10, C426C4, C4×Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C5×M4(2), C22×Dic5, C22×C20, C2×C4×Dic5, C23.21D10, C10×M4(2), C20.33C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4≀C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C426C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, D207C4, C10.10C42, C20.33C42

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D5 Dic5 D10 C4≀C2 Dic10 C4×D5 C5⋊D4 D20 D20⋊7C4 kernel C20.33C42 C2×C4×Dic5 C23.21D10 C10×M4(2) C4×Dic5 C4⋊Dic5 C5×M4(2) C2×C20 C2×C20 C22×C10 C2×M4(2) M4(2) C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 2 4 2 8 4 8 8 4 8

Matrix representation of C20.33C42 in GL4(𝔽41) generated by

 35 5 0 0 1 40 0 0 0 0 32 0 0 0 40 9
,
 28 4 0 0 19 13 0 0 0 0 1 23 0 0 0 40
,
 32 0 0 0 0 32 0 0 0 0 32 39 0 0 36 9
`G:=sub<GL(4,GF(41))| [35,1,0,0,5,40,0,0,0,0,32,40,0,0,0,9],[28,19,0,0,4,13,0,0,0,0,1,0,0,0,23,40],[32,0,0,0,0,32,0,0,0,0,32,36,0,0,39,9] >;`

C20.33C42 in GAP, Magma, Sage, TeX

`C_{20}._{33}C_4^2`
`% in TeX`

`G:=Group("C20.33C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,113);`
`// by ID`

`G=gap.SmallGroup(320,113);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^5*b>;`
`// generators/relations`

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