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## G = C42.3F5order 320 = 26·5

### 3rd non-split extension by C42 of F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C42.3F5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C42.3F5
 Lower central C5 — C20 — C42.3F5
 Upper central C1 — C4 — C42

Generators and relations for C42.3F5
G = < a,b,c,d | a4=b4=c5=1, d4=b, ab=ba, ac=ca, dad-1=ab-1, bc=cb, bd=db, dcd-1=c3 >

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C4 M4(2) M4(2) F5 C2×F5 C16⋊C4 C4.F5 C4×F5 C22.F5 C42.3F5 kernel C42.3F5 C42.D5 C20.C8 C5⋊C16 C2×C5⋊2C8 C4×C20 C20 C2×C10 C42 C2×C4 C5 C4 C4 C22 C1 # reps 1 1 2 8 2 2 2 2 1 1 2 2 2 2 8

Matrix representation of C42.3F5 in GL4(𝔽241) generated by

 1 0 0 0 0 240 0 0 0 0 64 0 0 0 0 177
,
 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 91 0 0 0 0 98 0 0 0 0 87 0 0 0 0 205
,
 0 0 1 0 0 0 0 1 0 1 0 0 64 0 0 0
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,240,0,0,0,0,64,0,0,0,0,177],[64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[91,0,0,0,0,98,0,0,0,0,87,0,0,0,0,205],[0,0,0,64,0,0,1,0,1,0,0,0,0,1,0,0] >;`

C42.3F5 in GAP, Magma, Sage, TeX

`C_4^2._3F_5`
`% in TeX`

`G:=Group("C4^2.3F5");`
`// GroupNames label`

`G:=SmallGroup(320,198);`
`// by ID`

`G=gap.SmallGroup(320,198);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,100,1123,102,6278,3156]);`
`// 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*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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