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G = M4(2)  order 16 = 24

Modular maximal-cyclic group

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: M4(2), C4.C4, C8⋊3C2, C22.C4, C4.6C22, C2.3(C2×C4), (C2×C4).2C2, 2-Sylow(AGammaL(1,25)), SmallGroup(16,6)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — M4(2)
Chief series
C1 — C2 — C4 — C2×C4 — M4(2)
Lower central
C1 — C2 — M4(2)
Upper central
C1 — C4 — M4(2)
Jennings
C1 — C2 — C2 — C4 — M4(2)

Generators and relations for M4(2)
 G = < a,b | a8=b2=1, bab=a5 >

C1
C2
2C2
C4
C22
C4
C8
C8
C2×C4
M4(2)

Character table of M4(2)

 class 12A2B4A4B4C8A8B8C8D
 size 1121122222
ρ11111111111    trivial
ρ211-111-11-1-11    linear of order 2
ρ311-111-1-111-1    linear of order 2
ρ4111111-1-1-1-1    linear of order 2
ρ511-1-1-11i-ii-i    linear of order 4
ρ6111-1-1-1ii-i-i    linear of order 4
ρ711-1-1-11-ii-ii    linear of order 4
ρ8111-1-1-1-i-iii    linear of order 4
ρ92-202i-2i00000    complex faithful
ρ102-20-2i2i00000    complex faithful

Permutation representations of M4(2)
►On 8 points - transitive group 8T7
Generators in S8
(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);
 

►Regular action on 16 points - transitive group 16T6
Generators in S16
(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  C4≀C2  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: C8○D4  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) ...

Polynomial with Galois group M4(2) over ℚ
actionf(x)Disc(f)
8T7x8-10x6+25x4-20x2+5216·57

Matrix representation of M4(2) ►in GL2(𝔽5) generated by

02
10
,
40
01
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

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