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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, C83C2, 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

C1C2 — M4(2)
C1C2C4C2×C4 — M4(2)
C1C2 — M4(2)
C1C4 — M4(2)
C1C2C2C4 — M4(2)

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

2C2

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);

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

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