M5(2):C2 - GroupNames

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G = M₅(2)⋊C₂ order 64 = 2⁶

6^th semidirect product of M₅(2) and C₂ acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D₈.₂C₄, C₄.₁₁D₈, C₈.₁₆D₄, M₅(2)⋊₆C₂, C₂².₃SD₁₆, C₈.₂(C₂×C₄), (C₂×D₈).₅C₂, (C₂×C₄).₁₁D₄, C₈.C₄⋊₂C₂, C₄.₅(C₂²⋊C₄), (C₂×C₈).₁₀C₂², C₂.₁₀(D₄⋊C₄), 2-Sylow(PSigmaU(2,81)), SmallGroup(64,42)

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

C₁ — C₈ — M₅(2)⋊C₂

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂×D₈ — M₅(2)⋊C₂

C₁ — C₂ — C₄ — C₈ — M₅(2)⋊C₂

C₁ — C₂ — C₂×C₄ — C₂×C₈ — M₅(2)⋊C₂

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — M₅(2)⋊C₂

Generators and relations for M₅(2)⋊C₂
G = < a,b,c | a¹⁶=b²=c²=1, bab=a⁹, cac=a³b, bc=cb >

C₁

C₂

₂C₂

₈C₂

₈C₂

C₂²

C₄

C₄

₄C₂²

₄C₂²

₈C₂²

₈C₂²

C₈

C₈

₂D₄

₂D₄

₄D₄

₄C₂³

₄C₈

D₈

D₈

₂C₁₆

₂M₄(2)

₂D₈

₂C₂×D₄

Character table of M₅(2)⋊C₂

class	1	2A	2B	2C	2D	4A	4B	8A	8B	8C	8D	8E	16A	16B	16C	16D
size	1	1	2	8	8	2	2	2	2	4	8	8	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	-1	1	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	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	1	-1	1	-1	-1	1	i	-i	-i	i	-i	i	linear of order 4
ρ₆	1	1	-1	1	-1	-1	1	-1	-1	1	-i	i	-i	i	-i	i	linear of order 4
ρ₇	1	1	-1	1	-1	-1	1	-1	-1	1	i	-i	i	-i	i	-i	linear of order 4
ρ₈	1	1	-1	-1	1	-1	1	-1	-1	1	-i	i	i	-i	i	-i	linear of order 4
ρ₉	2	2	-2	0	0	-2	2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	0	2	2	-2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	0	0	2	-2	0	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	-2	0	0	2	-2	0	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	0	0	-2	-2	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₁₄	2	2	2	0	0	-2	-2	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₅	4	-4	0	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₆	4	-4	0	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of M₅(2)⋊C₂
►On 16 points - transitive group 16T144

Generators in S₁₆

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(1 9)(3 11)(5 13)(7 15)

(2 4)(3 15)(5 13)(6 16)(7 11)(8 14)(10 12)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,9)(3,11)(5,13)(7,15), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,9)(3,11)(5,13)(7,15), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,9),(3,11),(5,13),(7,15)], [(2,4),(3,15),(5,13),(6,16),(7,11),(8,14),(10,12)]])

G:=TransitiveGroup(16,144);

M₅(2)⋊C₂ is a maximal subgroup of
C₂³.₂₁SD₁₆ D₁₆⋊C₄ Q₁₆.₁₀D₄ D₈.₃D₄ D₄.₃D₈ D₄.₅D₈ M₅(2).C₂² D₄₀.C₄
C_4p.D₈: C₈.₃D₈ D₈⋊₃D₄ D₂₄.C₄ M₅(2)⋊S₃ D₈.Dic₃ D₄₀.₆C₄ D₄₀.₄C₄ D₈.Dic₅ ...
M₅(2)⋊C₂ is a maximal quotient of
D₈⋊C₈ C₂³.₁₂SD₁₆ C₄.₆Q₃₂ C₈.₂C₄² D₄₀.C₄
C_4p.D₈: C₄.D₁₆ D₂₄.C₄ M₅(2)⋊S₃ D₈.Dic₃ D₄₀.₆C₄ D₄₀.₄C₄ D₈.Dic₅ D₅₆.C₄ ...

Matrix representation of M₅(2)⋊C₂ ►in GL₄(𝔽₇) generated by

2	3	4	0
0	6	4	0
2	1	5	2
4	5	0	1

,

1	0	6	3
0	2	3	2
0	3	2	2
0	1	1	2

,

1	6	1	0
0	3	6	5
0	4	5	5
0	2	3	5

G:=sub<GL(4,GF(7))| [2,0,2,4,3,6,1,5,4,4,5,0,0,0,2,1],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2],[1,0,0,0,6,3,4,2,1,6,5,3,0,5,5,5] >;

M₅(2)⋊C₂ in GAP, Magma, Sage, TeX

M_5(2)\rtimes C_2

% in TeX

G:=Group("M5(2):C2");

// GroupNames label

G:=SmallGroup(64,42);

// by ID

G=gap.SmallGroup(64,42);

# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,476,86,489,117,1444,730,88]);

// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^2=1,b*a*b=a^9,c*a*c=a^3*b,b*c=c*b>;

// generators/relations

Export

Subgroup lattice of M₅(2)⋊C₂ in TeX
Character table of M₅(2)⋊C₂ in TeX

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