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## G = C20.C8order 160 = 25·5

### 1st non-split extension by C20 of C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C20.C8
 Lower central C5 — C10 — C20.C8
 Upper central C1 — C4 — C2×C4

Generators and relations for C20.C8
G = < a,b | a20=1, b8=a10, bab-1=a3 >

Character table of C20.C8

 class 1 2A 2B 4A 4B 4C 5 8A 8B 8C 8D 8E 8F 10A 10B 10C 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D size 1 1 2 1 1 2 4 5 5 5 5 10 10 4 4 4 10 10 10 10 10 10 10 10 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 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 -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 -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 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 i -i -i i i -i -i i 1 1 1 1 linear of order 4 ρ6 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 i i i -i -i -i -i i 1 1 -1 -1 linear of order 4 ρ7 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -i i i -i -i i i -i 1 1 1 1 linear of order 4 ρ8 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 -i -i -i i i i i -i 1 1 -1 -1 linear of order 4 ρ9 1 1 -1 -1 -1 1 1 i i -i -i -i i -1 1 -1 ζ85 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 -1 -1 1 1 linear of order 8 ρ10 1 1 1 -1 -1 -1 1 i i -i -i i -i 1 1 1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 -1 -1 -1 -1 linear of order 8 ρ11 1 1 -1 -1 -1 1 1 -i -i i i i -i -1 1 -1 ζ87 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 -1 -1 1 1 linear of order 8 ρ12 1 1 1 -1 -1 -1 1 -i -i i i -i i 1 1 1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 -1 -1 -1 -1 linear of order 8 ρ13 1 1 1 -1 -1 -1 1 -i -i i i -i i 1 1 1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 -1 -1 -1 -1 linear of order 8 ρ14 1 1 -1 -1 -1 1 1 i i -i -i -i i -1 1 -1 ζ8 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 -1 -1 1 1 linear of order 8 ρ15 1 1 1 -1 -1 -1 1 i i -i -i i -i 1 1 1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 -1 -1 -1 -1 linear of order 8 ρ16 1 1 -1 -1 -1 1 1 -i -i i i i -i -1 1 -1 ζ83 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 -1 -1 1 1 linear of order 8 ρ17 2 -2 0 -2i 2i 0 2 2ζ83 2ζ87 2ζ85 2ζ8 0 0 0 -2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from M5(2) ρ18 2 -2 0 2i -2i 0 2 2ζ85 2ζ8 2ζ83 2ζ87 0 0 0 -2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from M5(2) ρ19 2 -2 0 2i -2i 0 2 2ζ8 2ζ85 2ζ87 2ζ83 0 0 0 -2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 complex lifted from M5(2) ρ20 2 -2 0 -2i 2i 0 2 2ζ87 2ζ83 2ζ8 2ζ85 0 0 0 -2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 complex lifted from M5(2) ρ21 4 4 -4 4 4 -4 -1 0 0 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 4 4 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 -4 -4 -4 4 -1 0 0 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 1 1 -1 -1 symplectic lifted from C5⋊C8, Schur index 2 ρ24 4 4 4 -4 -4 -4 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 symplectic lifted from C5⋊C8, Schur index 2 ρ25 4 -4 0 -4i 4i 0 -1 0 0 0 0 0 0 √5 1 -√5 0 0 0 0 0 0 0 0 -i i √-5 -√-5 complex faithful, Schur index 2 ρ26 4 -4 0 4i -4i 0 -1 0 0 0 0 0 0 √5 1 -√5 0 0 0 0 0 0 0 0 i -i -√-5 √-5 complex faithful, Schur index 2 ρ27 4 -4 0 -4i 4i 0 -1 0 0 0 0 0 0 -√5 1 √5 0 0 0 0 0 0 0 0 -i i -√-5 √-5 complex faithful, Schur index 2 ρ28 4 -4 0 4i -4i 0 -1 0 0 0 0 0 0 -√5 1 √5 0 0 0 0 0 0 0 0 i -i √-5 -√-5 complex faithful, Schur index 2

Smallest permutation representation of C20.C8
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 74 21 48 6 69 26 43 11 64 31 58 16 79 36 53)(2 61 30 51 7 76 35 46 12 71 40 41 17 66 25 56)(3 68 39 54 8 63 24 49 13 78 29 44 18 73 34 59)(4 75 28 57 9 70 33 52 14 65 38 47 19 80 23 42)(5 62 37 60 10 77 22 55 15 72 27 50 20 67 32 45)

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

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

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

C20.C8 is a maximal subgroup of
C42.3F5  C42.9F5  C40.1C8  C20.23C42  C20.10M4(2)  D20.C8  C20.29M4(2)  D4.(C5⋊C8)  D5⋊M5(2)  Dic10.C8  C5⋊C16.C22  C15⋊M5(2)  C60.C8
C20.C8 is a maximal quotient of
C20⋊C16  C42.4F5  C10.6M5(2)  C15⋊M5(2)  C60.C8

Matrix representation of C20.C8 in GL6(𝔽241)

 177 0 0 0 0 0 78 64 0 0 0 0 0 0 0 0 0 1 0 0 240 0 0 1 0 0 0 240 0 1 0 0 0 0 240 1
,
 172 239 0 0 0 0 76 69 0 0 0 0 0 0 197 177 10 201 0 0 207 137 148 157 0 0 104 93 84 167 0 0 40 103 44 64

G:=sub<GL(6,GF(241))| [177,78,0,0,0,0,0,64,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,1,1,1],[172,76,0,0,0,0,239,69,0,0,0,0,0,0,197,207,104,40,0,0,177,137,93,103,0,0,10,148,84,44,0,0,201,157,167,64] >;

C20.C8 in GAP, Magma, Sage, TeX

C_{20}.C_8
% in TeX

G:=Group("C20.C8");
// GroupNames label

G:=SmallGroup(160,73);
// by ID

G=gap.SmallGroup(160,73);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,50,69,2309,1169]);
// Polycyclic

G:=Group<a,b|a^20=1,b^8=a^10,b*a*b^-1=a^3>;
// generators/relations

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