p-group, metacyclic, nilpotent (class 2), monomial
Aliases: M4(2), C4.C4, C8:3C2, C22.C4, C4.6C22, C2.3(C2xC4), (C2xC4).2C2, 2-Sylow(AGammaL(1,25)), SmallGroup(16,6)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)
G = < a,b | a8=b2=1, bab=a5 >
Character table of M4(2)
class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | |
size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 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 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | linear of order 4 |
ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | linear of order 4 |
ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
ρ9 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ10 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
(1 2 3 4 5 6 7 8)
(2 6)(4 8)
G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (2,6)(4,8)>;
G:=Group( (1,2,3,4,5,6,7,8), (2,6)(4,8) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8)], [(2,6),(4,8)]])
G:=TransitiveGroup(8,7);
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,11),(3,16),(4,13),(5,10),(6,15),(7,12),(8,9)]])
G:=TransitiveGroup(16,6);
M4(2) is a maximal subgroup of
C4.D4 C4.10D4 C4wrC2 C8:C22 C8.C22 C32:M4(2) C62.C4 C52:M4(2)
C4p.C4: C8.C4 C4.Dic3 C4.Dic5 C4.F5 C4.Dic7 C44.C4 C52.4C4 C52.C4 ...
D2p.C4: C8oD4 C8:S3 C8:D5 C8:D7 C88:C2 C8:D13 C8:D17 C8:D19 ...
Cp:M4(2), p=1 mod 4: C22.F5 C13:M4(2) C17:M4(2) C29:M4(2) ...
M4(2) is a maximal quotient of
C8:C4 C32:M4(2) C62.C4 C52:M4(2)
C4.D2p: C22:C8 C4:C8 C8:S3 C4.Dic3 C8:D5 C4.Dic5 C8:D7 C4.Dic7 ...
Cp:M4(2), p=1 mod 4: C4.F5 C22.F5 C52.C4 C13:M4(2) D34.4C4 C17:M4(2) C116.C4 C29:M4(2) ...
action | f(x) | Disc(f) |
---|---|---|
8T7 | x8-10x6+25x4-20x2+5 | 216·57 |
Matrix representation of M4(2) ►in GL2(F5) generated by
0 | 2 |
1 | 0 |
4 | 0 |
0 | 1 |
G:=sub<GL(2,GF(5))| [0,1,2,0],[4,0,0,1] >;
M4(2) in GAP, Magma, Sage, TeX
M_4(2)
% in TeX
G:=Group("M4(2)");
// GroupNames label
G:=SmallGroup(16,6);
// by ID
G=gap.SmallGroup(16,6);
# by ID
G:=PCGroup([4,-2,2,-2,-2,16,81,34]);
// Polycyclic
G:=Group<a,b|a^8=b^2=1,b*a*b=a^5>;
// generators/relations
Export
Subgroup lattice of M4(2) in TeX
Character table of M4(2) in TeX