direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8xF5, Dic10:3C4, D10.13C23, C5:(C4xQ8), (C5xQ8):3C4, C4:F5.2C2, C4.7(C2xF5), C20.7(C2xC4), (C4xF5).1C2, D5.2(C2xQ8), (Q8xD5).3C2, D5.3(C4oD4), Dic5.2(C2xC4), (C2xF5).4C22, C2.11(C22xF5), C10.10(C22xC4), (C4xD5).14C22, SmallGroup(160,209)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8xF5
G = < a,b,c,d | a4=c5=d4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 204 in 70 conjugacy classes, 38 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2xC4, Q8, Q8, D5, C10, C42, C4:C4, C2xQ8, Dic5, C20, F5, F5, D10, C4xQ8, Dic10, C4xD5, C5xQ8, C2xF5, C2xF5, C4xF5, C4:F5, Q8xD5, Q8xF5
Quotients: C1, C2, C4, C22, C2xC4, Q8, C23, C22xC4, C2xQ8, C4oD4, F5, C4xQ8, C2xF5, C22xF5, Q8xF5
Character table of Q8xF5
class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5 | 10 | 20A | 20B | 20C | |
size | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 8 | 8 | 8 | |
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 | 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 | 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 | 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 | linear of order 2 |
ρ5 | 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 |
ρ6 | 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 |
ρ7 | 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 |
ρ8 | 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 |
ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -i | -1 | -1 | -1 | i | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ10 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | i | 1 | 1 | -1 | i | -i | -i | i | -i | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | i | -i | -i | i | -i | -1 | 1 | 1 | i | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
ρ12 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | -i | i | i | 1 | -1 | 1 | i | -i | i | -i | -i | 1 | 1 | -1 | 1 | -1 | linear of order 4 |
ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | i | i | -i | i | -1 | 1 | 1 | -i | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | i | i | -i | -i | 1 | -1 | 1 | -i | i | -i | i | i | 1 | 1 | -1 | 1 | -1 | linear of order 4 |
ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | i | -1 | -1 | -1 | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ16 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | -i | 1 | 1 | -1 | -i | i | i | -i | i | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ21 | 4 | 4 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from C2xF5 |
ρ22 | 4 | 4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
ρ23 | 4 | 4 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2xF5 |
ρ24 | 4 | 4 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from C2xF5 |
ρ25 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(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)
(1 21 6 26)(2 23 10 29)(3 25 9 27)(4 22 8 30)(5 24 7 28)(11 36 16 31)(12 38 20 34)(13 40 19 32)(14 37 18 35)(15 39 17 33)
G:=sub<Sym(40)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (1,21,6,26)(2,23,10,29)(3,25,9,27)(4,22,8,30)(5,24,7,28)(11,36,16,31)(12,38,20,34)(13,40,19,32)(14,37,18,35)(15,39,17,33)>;
G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35), (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), (1,21,6,26)(2,23,10,29)(3,25,9,27)(4,22,8,30)(5,24,7,28)(11,36,16,31)(12,38,20,34)(13,40,19,32)(14,37,18,35)(15,39,17,33) );
G=PermutationGroup([[(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,26,6,21),(2,27,7,22),(3,28,8,23),(4,29,9,24),(5,30,10,25),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35)], [(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)], [(1,21,6,26),(2,23,10,29),(3,25,9,27),(4,22,8,30),(5,24,7,28),(11,36,16,31),(12,38,20,34),(13,40,19,32),(14,37,18,35),(15,39,17,33)]])
Q8xF5 is a maximal subgroup of
SD16:F5 Dic20:C4 D5.2- 1+4 D5.2+ 1+4 Dic6:5F5
Q8xF5 is a maximal quotient of
Dic10:C8 Dic5.M4(2) C20.M4(2) C4:C4:5F5 C20:(C4:C4) C20.6M4(2) (C2xF5):Q8 Dic6:5F5
Matrix representation of Q8xF5 ►in GL6(F41)
0 | 1 | 0 | 0 | 0 | 0 |
40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
9 | 0 | 0 | 0 | 0 | 0 |
0 | 32 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 1 | 0 | 0 | 40 |
0 | 0 | 0 | 1 | 0 | 40 |
0 | 0 | 0 | 0 | 1 | 40 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;
Q8xF5 in GAP, Magma, Sage, TeX
Q_8\times F_5
% in TeX
G:=Group("Q8xF5");
// GroupNames label
G:=SmallGroup(160,209);
// by ID
G=gap.SmallGroup(160,209);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,2309,599]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^5=d^4=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export