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## G = C42.9F5order 320 = 26·5

### 6th non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C42.9F5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C42.9F5
 Lower central C5 — C10 — C20 — C42.9F5
 Upper central C1 — C4 — C2×C4 — C42

Generators and relations for C42.9F5
G = < a,b,c,d | a4=b4=c5=1, d4=b, ab=ba, ac=ca, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of C42.9F5
On 80 points
Generators in S80
```(2 14 10 6)(4 16 12 8)(17 29 25 21)(19 31 27 23)(34 46 42 38)(36 48 44 40)(49 61 57 53)(51 63 59 55)(65 77 73 69)(67 79 75 71)
(1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)(33 37 41 45)(34 38 42 46)(35 39 43 47)(36 40 44 48)(49 53 57 61)(50 54 58 62)(51 55 59 63)(52 56 60 64)(65 69 73 77)(66 70 74 78)(67 71 75 79)(68 72 76 80)
(1 28 70 43 54)(2 44 29 55 71)(3 56 45 72 30)(4 73 57 31 46)(5 32 74 47 58)(6 48 17 59 75)(7 60 33 76 18)(8 77 61 19 34)(9 20 78 35 62)(10 36 21 63 79)(11 64 37 80 22)(12 65 49 23 38)(13 24 66 39 50)(14 40 25 51 67)(15 52 41 68 26)(16 69 53 27 42)
(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)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 4A 4B 4C ··· 4G 5 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C 16A ··· 16H 20A ··· 20L order 1 2 2 4 4 4 ··· 4 5 8 8 8 8 8 ··· 8 10 10 10 16 ··· 16 20 ··· 20 size 1 1 2 1 1 2 ··· 2 4 5 5 5 5 10 ··· 10 4 4 4 20 ··· 20 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + + - + + - image C1 C2 C2 C4 C4 C8 D4 Q8 M4(2) C8.C8 F5 C2×F5 D5⋊C8 C4⋊F5 C22.F5 C42.9F5 kernel C42.9F5 C4×C5⋊2C8 C20.C8 C2×C5⋊2C8 C4×C20 C5⋊2C8 C5⋊2C8 C5⋊2C8 C2×C10 C5 C42 C2×C4 C4 C4 C22 C1 # reps 1 1 2 2 2 8 1 1 2 8 1 1 2 2 2 8

Matrix representation of C42.9F5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 177 0 0 0 0 177
,
 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 190 240 0 0 191 240 0 0 0 0 0 52 0 0 190 51
,
 0 0 1 0 0 0 0 1 140 127 0 0 146 101 0 0
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,177,0,0,0,0,177],[64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[190,191,0,0,240,240,0,0,0,0,0,190,0,0,52,51],[0,0,140,146,0,0,127,101,1,0,0,0,0,1,0,0] >;`

C42.9F5 in GAP, Magma, Sage, TeX

`C_4^2._9F_5`
`% in TeX`

`G:=Group("C4^2.9F5");`
`// GroupNames label`

`G:=SmallGroup(320,199);`
`// by ID`

`G=gap.SmallGroup(320,199);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,100,1123,136,102,6278,3156]);`
`// Polycyclic`

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

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