metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8.3F5, C40.5C4, C5⋊1M5(2), D10.3C8, Dic5.3C8, C5⋊C16⋊1C2, C10.2(C2×C8), (C4×D5).6C4, (C8×D5).8C2, C4.15(C2×F5), C20.14(C2×C4), C2.3(D5⋊C8), C5⋊2C8.15C22, SmallGroup(160,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.F5
G = < a,b,c | a8=b5=1, c4=a2, ab=ba, cac-1=a5, cbc-1=b3 >
Character table of C8.F5
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 4 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 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 | i | i | -i | -i | -i | -i | i | i | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 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 | -1 | -1 | linear of order 4 |
| ρ7 | 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 | 1 | 1 | linear of order 4 |
| ρ8 | 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 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | -i | 1 | ζ87 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ83 | -1 | -1 | i | -i | i | -i | linear of order 8 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | -i | 1 | ζ83 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ87 | -1 | -1 | i | -i | i | -i | linear of order 8 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | 1 | ζ85 | ζ8 | ζ87 | ζ83 | ζ87 | ζ83 | ζ8 | ζ85 | -1 | -1 | i | -i | i | -i | linear of order 8 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | 1 | ζ8 | ζ85 | ζ83 | ζ87 | ζ83 | ζ87 | ζ85 | ζ8 | -1 | -1 | i | -i | i | -i | linear of order 8 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | i | 1 | ζ85 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ8 | -1 | -1 | -i | i | -i | i | linear of order 8 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | 1 | ζ83 | ζ87 | ζ8 | ζ85 | ζ8 | ζ85 | ζ87 | ζ83 | -1 | -1 | -i | i | -i | i | linear of order 8 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | 1 | ζ87 | ζ83 | ζ85 | ζ8 | ζ85 | ζ8 | ζ83 | ζ87 | -1 | -1 | -i | i | -i | i | linear of order 8 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | i | 1 | ζ8 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ85 | -1 | -1 | -i | i | -i | i | linear of order 8 |
| ρ17 | 2 | -2 | 0 | -2i | 2i | 0 | 2 | 0 | 0 | 2ζ8 | 2ζ87 | 2ζ85 | 2ζ83 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ18 | 2 | -2 | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 2ζ87 | 2ζ8 | 2ζ83 | 2ζ85 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ19 | 2 | -2 | 0 | 2i | -2i | 0 | 2 | 0 | 0 | 2ζ83 | 2ζ85 | 2ζ87 | 2ζ8 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ20 | 2 | -2 | 0 | -2i | 2i | 0 | 2 | 0 | 0 | 2ζ85 | 2ζ83 | 2ζ8 | 2ζ87 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ21 | 4 | 4 | 0 | 4 | 4 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ22 | 4 | 4 | 0 | 4 | 4 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 0 | -4 | -4 | 0 | -1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | i | -i | i | -i | complex lifted from D5⋊C8, Schur index 2 |
| ρ24 | 4 | 4 | 0 | -4 | -4 | 0 | -1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -i | i | -i | i | complex lifted from D5⋊C8, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 4i | -4i | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ83ζ53+2ζ83ζ52+ζ83 | complex faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | -4i | 4i | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ8ζ54+2ζ8ζ5+ζ8 | complex faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 4i | -4i | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ85ζ54+2ζ85ζ5+ζ85 | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ83ζ54+2ζ83ζ5+ζ83 | complex faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | -4i | 4i | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ83ζ54+2ζ83ζ5+ζ83 | 2ζ8ζ54+2ζ8ζ5+ζ8 | 2ζ83ζ53+2ζ83ζ52+ζ83 | 2ζ85ζ54+2ζ85ζ5+ζ85 | complex faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 11 5 15 9 3 13 7)(2 4 6 8 10 12 14 16)(17 62 21 50 25 54 29 58)(18 55 22 59 26 63 30 51)(19 64 23 52 27 56 31 60)(20 57 24 61 28 49 32 53)(33 70 37 74 41 78 45 66)(34 79 38 67 42 71 46 75)(35 72 39 76 43 80 47 68)(36 65 40 69 44 73 48 77)
(1 36 23 58 71)(2 59 37 72 24)(3 73 60 25 38)(4 26 74 39 61)(5 40 27 62 75)(6 63 41 76 28)(7 77 64 29 42)(8 30 78 43 49)(9 44 31 50 79)(10 51 45 80 32)(11 65 52 17 46)(12 18 66 47 53)(13 48 19 54 67)(14 55 33 68 20)(15 69 56 21 34)(16 22 70 35 57)
(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)| (1,11,5,15,9,3,13,7)(2,4,6,8,10,12,14,16)(17,62,21,50,25,54,29,58)(18,55,22,59,26,63,30,51)(19,64,23,52,27,56,31,60)(20,57,24,61,28,49,32,53)(33,70,37,74,41,78,45,66)(34,79,38,67,42,71,46,75)(35,72,39,76,43,80,47,68)(36,65,40,69,44,73,48,77), (1,36,23,58,71)(2,59,37,72,24)(3,73,60,25,38)(4,26,74,39,61)(5,40,27,62,75)(6,63,41,76,28)(7,77,64,29,42)(8,30,78,43,49)(9,44,31,50,79)(10,51,45,80,32)(11,65,52,17,46)(12,18,66,47,53)(13,48,19,54,67)(14,55,33,68,20)(15,69,56,21,34)(16,22,70,35,57), (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( (1,11,5,15,9,3,13,7)(2,4,6,8,10,12,14,16)(17,62,21,50,25,54,29,58)(18,55,22,59,26,63,30,51)(19,64,23,52,27,56,31,60)(20,57,24,61,28,49,32,53)(33,70,37,74,41,78,45,66)(34,79,38,67,42,71,46,75)(35,72,39,76,43,80,47,68)(36,65,40,69,44,73,48,77), (1,36,23,58,71)(2,59,37,72,24)(3,73,60,25,38)(4,26,74,39,61)(5,40,27,62,75)(6,63,41,76,28)(7,77,64,29,42)(8,30,78,43,49)(9,44,31,50,79)(10,51,45,80,32)(11,65,52,17,46)(12,18,66,47,53)(13,48,19,54,67)(14,55,33,68,20)(15,69,56,21,34)(16,22,70,35,57), (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([[(1,11,5,15,9,3,13,7),(2,4,6,8,10,12,14,16),(17,62,21,50,25,54,29,58),(18,55,22,59,26,63,30,51),(19,64,23,52,27,56,31,60),(20,57,24,61,28,49,32,53),(33,70,37,74,41,78,45,66),(34,79,38,67,42,71,46,75),(35,72,39,76,43,80,47,68),(36,65,40,69,44,73,48,77)], [(1,36,23,58,71),(2,59,37,72,24),(3,73,60,25,38),(4,26,74,39,61),(5,40,27,62,75),(6,63,41,76,28),(7,77,64,29,42),(8,30,78,43,49),(9,44,31,50,79),(10,51,45,80,32),(11,65,52,17,46),(12,18,66,47,53),(13,48,19,54,67),(14,55,33,68,20),(15,69,56,21,34),(16,22,70,35,57)], [(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)]])
C8.F5 is a maximal subgroup of
C16⋊F5 C16⋊4F5 D10.D8 D40.C4 D40⋊1C4 Dic20.C4 D5⋊M5(2) Dic10.C8 D30.C8 C120.C4
C8.F5 is a maximal quotient of
C40.C8 D10⋊C16 C10.M5(2) D30.C8 C120.C4
Matrix representation of C8.F5 ►in GL6(𝔽241)
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 182 | 59 | 33 | 0 |
| 0 | 0 | 215 | 59 | 0 | 182 |
| 0 | 0 | 182 | 0 | 59 | 215 |
| 0 | 0 | 0 | 33 | 59 | 182 |
G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,30,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[0,30,0,0,0,0,1,0,0,0,0,0,0,0,182,215,182,0,0,0,59,59,0,33,0,0,33,0,59,59,0,0,0,182,215,182] >;
C8.F5 in GAP, Magma, Sage, TeX
C_8.F_5
% in TeX
G:=Group("C8.F5"); // GroupNames label
G:=SmallGroup(160,65);
// by ID
G=gap.SmallGroup(160,65);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,50,69,2309,1169]);
// Polycyclic
G:=Group<a,b,c|a^8=b^5=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C8.F5 in TeX
Character table of C8.F5 in TeX