p-group, metacyclic, nilpotent (class 2), monomial
Aliases: M5(2), C4.C8, C16⋊3C2, C8.2C4, C22.C8, C8.8C22, (C2×C8).8C2, (C2×C4).5C4, C2.3(C2×C8), C4.12(C2×C4), SmallGroup(32,17)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2)
G = < a,b | a16=b2=1, bab=a9 >
Character table of M5(2)
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | ζ85 | ζ85 | ζ87 | ζ87 | ζ83 | ζ83 | ζ8 | ζ8 | linear of order 8 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | ζ87 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | linear of order 8 |
| ρ11 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | ζ8 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | linear of order 8 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | ζ83 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | linear of order 8 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | ζ83 | ζ83 | ζ8 | ζ8 | ζ85 | ζ85 | ζ87 | ζ87 | linear of order 8 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | ζ8 | ζ8 | ζ83 | ζ83 | ζ87 | ζ87 | ζ85 | ζ85 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | ζ85 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | linear of order 8 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | ζ87 | ζ87 | ζ85 | ζ85 | ζ8 | ζ8 | ζ83 | ζ83 | linear of order 8 |
| ρ17 | 2 | -2 | 0 | 2i | -2i | 0 | 2ζ8 | 2ζ83 | 2ζ85 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 2 | -2 | 0 | -2i | 2i | 0 | 2ζ83 | 2ζ8 | 2ζ87 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 2 | -2 | 0 | 2i | -2i | 0 | 2ζ85 | 2ζ87 | 2ζ8 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 2 | -2 | 0 | -2i | 2i | 0 | 2ζ87 | 2ζ85 | 2ζ83 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T22
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 10)(4 12)(6 14)(8 16)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,10)(4,12)(6,14)(8,16) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,10),(4,12),(6,14),(8,16)]])
G:=TransitiveGroup(16,22);
M5(2) is a maximal subgroup of
C16⋊C4 C23.C8 D8⋊2C4 M5(2)⋊C2 C8.17D4 C8.Q8 C16⋊C22 Q32⋊C2 C32⋊3M5(2) C62.4C8 C4.F9 C22.F9
C4p.C8: C8.C8 C12.C8 C20.4C8 C20.C8 C28.C8 C44.C8 C52.4C8 C52.C8 ...
D2p.C8: D4.C8 D4○C16 D6.C8 C80⋊C2 C8.F5 C16⋊D7 D22.C8 C208⋊C2 ...
M5(2) is a maximal quotient of
C8.F5 C32⋊3M5(2) C62.4C8 C4.F9 C22.F9 D26.C8
C4p.C8: C16⋊5C4 C12.C8 C20.4C8 C20.C8 C28.C8 C44.C8 C52.4C8 C52.C8 ...
C8.D2p: C22⋊C16 C4⋊C16 D6.C8 C80⋊C2 C16⋊D7 D22.C8 C208⋊C2 ...
Matrix representation of M5(2) ►in GL2(𝔽17) generated by
| 9 | 15 |
| 14 | 8 |
| 1 | 0 |
| 9 | 16 |
G:=sub<GL(2,GF(17))| [9,14,15,8],[1,9,0,16] >;
M5(2) in GAP, Magma, Sage, TeX
M_5(2)
% in TeX
G:=Group("M5(2)"); // GroupNames label
G:=SmallGroup(32,17);
// by ID
G=gap.SmallGroup(32,17);
# by ID
G:=PCGroup([5,-2,2,-2,-2,-2,20,181,42,58]);
// Polycyclic
G:=Group<a,b|a^16=b^2=1,b*a*b=a^9>;
// generators/relations
Export
Subgroup lattice of M5(2) in TeX
Character table of M5(2) in TeX