M6(2) - GroupNames

G = M₆(2) order 64 = 2⁶

Modular maximal-cyclic group

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

Aliases: M₆(2), C₄.C₁₆, C₃₂:₃C₂, C₈.₃C₈, C₁₆.₂C₄, C₂².C₁₆, C₁₆.₈C₂², (C₂xC₄).₅C₈, C₄.₁₃(C₂xC₈), (C₂xC₁₆).₈C₂, (C₂xC₈).₁₃C₄, C₂.₃(C₂xC₁₆), C₈.₂₃(C₂xC₄), 2-Sylow(AGammaL(1,289)), SmallGroup(64,51)

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

Derived series

C₁ — C₂ — M₆(2)

Chief series

C₁ — C₂ — C₄ — C₈ — C₁₆ — C₂xC₁₆ — M₆(2)

Lower central

C₁ — C₂ — M₆(2)

Upper central

C₁ — C₁₆ — M₆(2)

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₄ — C₄ — C₈ — C₈ — C₁₆ — M₆(2)

Generators and relations for M₆(2)
G = < a,b | a³²=b²=1, bab=a¹⁷ >

Subgroups: 17 in 16 conjugacy classes, 15 normal (13 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂xC₄, C₁₆, C₂xC₈, C₂xC₁₆, M₆(2)

C₁

C₂

₂C₂

C₄

C₂²

C₂xC₄

C₈

C₁₆

C₂xC₈

C₁₆

C₃₂

C₂xC₁₆

M₆(2)

Smallest permutation representation of M₆(2)
►On 32 points

Generators in S₃₂

(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)

(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)

G:=sub<Sym(32)| (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)>;

G:=Group( (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), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32) );

G=PermutationGroup([[(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)], [(2,18),(4,20),(6,22),(8,24),(10,26),(12,28),(14,30),(16,32)]])

M₆(2) is a maximal subgroup of
C₂³.C₁₆ D₁₆:₃C₄ M₆(2):C₂ C₁₆.₁₈D₄ C₈.C₁₆ C₈.Q₁₆ C₃₂:C₂² Q₆₄:C₂
C_8p.C₈: C₃₂:C₄ C₃:M₆(2) C₈₀.₉C₄ C₅:M₆(2) C₇:M₆(2) ...
D_2p.C₁₆: D₄.C₁₆ D₄oC₃₂ C₉₆:C₂ C₃₂:D₅ C₈₀.C₄ C₃₂:D₇ ...
M₆(2) is a maximal quotient of
C₈₀.C₄
C_8p.C₈: C₃₂:₅C₄ C₃:M₆(2) C₈₀.₉C₄ C₅:M₆(2) C₇:M₆(2) ...
C₁₆.D_2p: C₂²:C₃₂ C₄:C₃₂ C₉₆:C₂ C₃₂:D₅ C₃₂:D₇ ...

40 conjugacy classes

class	1	2A	2B	4A	4B	4C	8A	8B	8C	8D	8E	8F	16A	···	16H	16I	16J	16K	16L	32A	···	32P
order	1	2	2	4	4	4	8	8	8	8	8	8	16	···	16	16	16	16	16	32	···	32
size	1	1	2	1	1	2	1	1	1	1	2	2	1	···	1	2	2	2	2	2	···	2

40 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2
type	+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆	C₁₆	M₆(2)
kernel	M₆(2)	C₃₂	C₂xC₁₆	C₁₆	C₂xC₈	C₈	C₂xC₄	C₄	C₂²	C₁
# reps	1	2	1	2	2	4	4	8	8	8

Matrix representation of M₆(2) ►in GL₂(F₁₇) generated by

0	11
1	0

16	0
0	1

G:=sub<GL(2,GF(17))| [0,1,11,0],[16,0,0,1] >;

M₆(2) in GAP, Magma, Sage, TeX

M_6(2)

% in TeX

G:=Group("M6(2)");

// GroupNames label

G:=SmallGroup(64,51);

// by ID

G=gap.SmallGroup(64,51);

# by ID

G:=PCGroup([6,-2,2,-2,-2,-2,-2,24,409,50,69,88]);

// Polycyclic

G:=Group<a,b|a^32=b^2=1,b*a*b=a^17>;

// generators/relations

Export

Subgroup lattice of M₆(2) in TeX

G = M6(2) order 64 = 26

Modular maximal-cyclic group

G = M₆(2) order 64 = 2⁶