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G = M6(2)  order 64 = 26

Modular maximal-cyclic group

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

Aliases: M6(2), C4.C16, C323C2, C8.3C8, C16.2C4, C22.C16, C16.8C22, (C2×C4).5C8, C4.13(C2×C8), (C2×C16).8C2, (C2×C8).13C4, C2.3(C2×C16), C8.23(C2×C4), 2-Sylow(AGammaL(1,289)), SmallGroup(64,51)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — M6(2)
C1C2C4C8C16C2×C16 — M6(2)
C1C2 — M6(2)
C1C16 — M6(2)
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — M6(2)

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

2C2

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  D163C4  M6(2)⋊C2  C16.18D4  C8.C16  C8.Q16  C32⋊C22  Q64⋊C2
 C8p.C8: C32⋊C4  C3⋊M6(2)  C80.9C4  C5⋊M6(2)  C7⋊M6(2) ...
 D2p.C16: D4.C16  D4○C32  C96⋊C2  C32⋊D5  C80.C4  C32⋊D7 ...
M6(2) is a maximal quotient of
C80.C4
 C8p.C8: C325C4  C3⋊M6(2)  C80.9C4  C5⋊M6(2)  C7⋊M6(2) ...
 C16.D2p: C22⋊C32  C4⋊C32  C96⋊C2  C32⋊D5  C32⋊D7 ...

40 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D8E8F16A···16H16I16J16K16L32A···32P
order12244488888816···161616161632···32
size1121121111221···122222···2

40 irreducible representations

dim1111111112
type+++
imageC1C2C2C4C4C8C8C16C16M6(2)
kernelM6(2)C32C2×C16C16C2×C8C8C2×C4C4C22C1
# reps1212244888

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

011
10
,
160
01
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

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