metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20.1C8, C5⋊2M5(2), C4.(C5⋊C8), C5⋊C16⋊2C2, C22.(C5⋊C8), (C2×C4).5F5, C5⋊2C8.6C4, (C2×C20).7C4, C10.6(C2×C8), (C2×C10).2C8, C4.19(C2×F5), C20.18(C2×C4), C5⋊2C8.17C22, C2.3(C2×C5⋊C8), (C2×C5⋊2C8).11C2, SmallGroup(160,73)
Series: Derived ►Chief ►Lower central ►Upper central
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 |
►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
Export
Subgroup lattice of C20.C8 in TeX
Character table of C20.C8 in TeX