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## G = C40.C4order 160 = 25·5

### 2nd non-split extension by C40 of C4 acting faithfully

Aliases: C8.2F5, C40.2C4, D10.1Q8, Dic5.10D4, C52C8.3C4, (C8×D5).6C2, C2.6(C4⋊F5), C10.3(C4⋊C4), C4.10(C2×F5), C51(C8.C4), C4.F5.1C2, C20.10(C2×C4), (C4×D5).27C22, SmallGroup(160,70)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40.C4
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C40.C4
 Lower central C5 — C10 — C20 — C40.C4
 Upper central C1 — C2 — C4 — C8

Generators and relations for C40.C4
G = < a,b | a40=1, b4=a20, bab-1=a3 >

Character table of C40.C4

 class 1 2A 2B 4A 4B 4C 5 8A 8B 8C 8D 8E 8F 8G 8H 10 20A 20B 40A 40B 40C 40D size 1 1 10 2 5 5 4 2 2 10 10 20 20 20 20 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 -1 -1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 -1 -1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 0 -2i 2i 2 √-2 -√-2 -√2 √2 0 0 0 0 -2 0 0 -√-2 √-2 √-2 -√-2 complex lifted from C8.C4 ρ12 2 -2 0 0 2i -2i 2 √-2 -√-2 √2 -√2 0 0 0 0 -2 0 0 -√-2 √-2 √-2 -√-2 complex lifted from C8.C4 ρ13 2 -2 0 0 2i -2i 2 -√-2 √-2 -√2 √2 0 0 0 0 -2 0 0 √-2 -√-2 -√-2 √-2 complex lifted from C8.C4 ρ14 2 -2 0 0 -2i 2i 2 -√-2 √-2 √2 -√2 0 0 0 0 -2 0 0 √-2 -√-2 -√-2 √-2 complex lifted from C8.C4 ρ15 4 4 0 4 0 0 -1 -4 -4 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ16 4 4 0 4 0 0 -1 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 4 0 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4⋊F5 ρ18 4 4 0 -4 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4⋊F5 ρ19 4 -4 0 0 0 0 -1 -2√-2 2√-2 0 0 0 0 0 0 1 -√-5 √-5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 complex faithful ρ20 4 -4 0 0 0 0 -1 2√-2 -2√-2 0 0 0 0 0 0 1 √-5 -√-5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 complex faithful ρ21 4 -4 0 0 0 0 -1 -2√-2 2√-2 0 0 0 0 0 0 1 √-5 -√-5 ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 complex faithful ρ22 4 -4 0 0 0 0 -1 2√-2 -2√-2 0 0 0 0 0 0 1 -√-5 √-5 ζ83ζ53+ζ83ζ52+ζ83-ζ8ζ53-ζ8ζ52 -ζ87ζ54-ζ87ζ5+ζ85ζ54+ζ85ζ5+ζ85 ζ87ζ54+ζ87ζ5+ζ87-ζ85ζ54-ζ85ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52+ζ8 complex faithful

Smallest permutation representation of C40.C4
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 76 31 46 21 56 11 66)(2 63 40 49 22 43 20 69)(3 50 9 52 23 70 29 72)(4 77 18 55 24 57 38 75)(5 64 27 58 25 44 7 78)(6 51 36 61 26 71 16 41)(8 65 14 67 28 45 34 47)(10 79 32 73 30 59 12 53)(13 80 19 42 33 60 39 62)(15 54 37 48 35 74 17 68)```

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

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

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

C40.C4 is a maximal subgroup of
C804C4  C805C4  D40.C4  Dic20.C4  (C8×D5).C4  M4(2).1F5  D8⋊F5  SD163F5  Q16⋊F5  D10.Dic6  C40.Dic3
C40.C4 is a maximal quotient of
C402C8  Dic5.13D8  C20.10C42  D10.Dic6  C40.Dic3

Matrix representation of C40.C4 in GL4(𝔽3) generated by

 0 0 2 2 1 0 1 1 2 1 2 1 0 0 2 1
,
 0 1 1 0 2 0 2 1 1 0 2 1 1 1 1 1
`G:=sub<GL(4,GF(3))| [0,1,2,0,0,0,1,0,2,1,2,2,2,1,1,1],[0,2,1,1,1,0,0,1,1,2,2,1,0,1,1,1] >;`

C40.C4 in GAP, Magma, Sage, TeX

`C_{40}.C_4`
`% in TeX`

`G:=Group("C40.C4");`
`// GroupNames label`

`G:=SmallGroup(160,70);`
`// by ID`

`G=gap.SmallGroup(160,70);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,579,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b|a^40=1,b^4=a^20,b*a*b^-1=a^3>;`
`// generators/relations`

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