M4(2):4C4 - GroupNames

G = M₄(2):₄C₄ order 64 = 2⁶

4^th semidirect product of M₄(2) and C₄ acting via C₄/C₂=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M₄(2):₄C₄, C₂².₃C₄², (C₂xC₈):₂C₄, C₄.₅(C₄:C₄), (C₂xC₄).₃Q₈, (C₂xC₄).₁₁₄D₄, C₂²:C₄.₁C₄, C₂³.₆(C₂xC₄), C₂².₆(C₄:C₄), C₄.₂₂(C₂²:C₄), C₄²:C₂.₃C₂, (C₂xM₄(2)).₉C₂, C₂².₈(C₂²:C₄), (C₂²xC₄).₂₁C₂², C₂.₉(C₂.C₄²), (C₂xC₄).₆₅(C₂xC₄), SmallGroup(64,25)

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

Derived series

C₁ — C₂² — M₄(2):₄C₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂xM₄(2) — M₄(2):₄C₄

Lower central

C₁ — C₂ — C₂² — M₄(2):₄C₄

Upper central

C₁ — C₄ — C₂²xC₄ — M₄(2):₄C₄

Jennings

C₁ — C₂ — C₂ — C₂²xC₄ — M₄(2):₄C₄

Generators and relations for M₄(2):₄C₄
G = < a,b,c | a⁸=b²=c⁴=1, bab=a⁵, cac^-1=ab, cbc^-1=a⁴b >

Subgroups: 69 in 45 conjugacy classes, 27 normal (21 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, Q₈, C₄², C₂²:C₄, C₄:C₄, C₂.C₄², M₄(2):₄C₄

C₁

C₂

₂C₂

C₄

C₂²

C₄

₄C₄

₄C₂²

C₂xC₄

C₂³

C₂xC₄

₂C₈

₂C₂xC₄

M₄(2)

C₂xC₈

C₂²:C₄

C₂²xC₄

C₂²:C₄

₂M₄(2)

₂C₄²

₂C₂xC₈

₂M₄(2)

₂C₄:C₄

C₂xM₄(2)

C₄²:C₂

M₄(2):₄C₄

Character table of M₄(2):₄C₄

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	2	2	2	1	1	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	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	-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	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	-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	1	1	-i	i	i	-1	-1	linear of order 4
ρ₆	1	1	1	-1	-1	-1	-1	1	-1	1	i	-i	-i	i	1	i	-i	-1	-1	1	-i	i	linear of order 4
ρ₇	1	1	1	-1	-1	1	1	-1	1	-1	-i	i	-i	i	i	-1	-1	i	-i	-i	1	1	linear of order 4
ρ₈	1	1	1	-1	-1	-1	-1	1	-1	1	i	-i	-i	i	-1	-i	i	1	1	-1	i	-i	linear of order 4
ρ₉	1	1	1	-1	-1	1	1	-1	1	-1	i	-i	i	-i	-i	-1	-1	-i	i	i	1	1	linear of order 4
ρ₁₀	1	1	1	-1	-1	-1	-1	1	-1	1	-i	i	i	-i	1	-i	i	-1	-1	1	i	-i	linear of order 4
ρ₁₁	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-1	-1	i	-i	i	-i	i	-i	-i	i	linear of order 4
ρ₁₂	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	1	1	i	i	-i	-i	i	-i	i	-i	linear of order 4
ρ₁₃	1	1	1	-1	-1	1	1	-1	1	-1	i	-i	i	-i	i	1	1	i	-i	-i	-1	-1	linear of order 4
ρ₁₄	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-1	-1	-i	i	-i	i	-i	i	i	-i	linear of order 4
ρ₁₅	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	1	1	-i	-i	i	i	-i	i	-i	i	linear of order 4
ρ₁₆	1	1	1	-1	-1	-1	-1	1	-1	1	-i	i	i	-i	-1	i	-i	1	1	-1	-i	i	linear of order 4
ρ₁₇	2	2	-2	-2	2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	-2	-2	2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	-2	2	-2	2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	-2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	4	-4	0	0	0	4i	-4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₂	4	-4	0	0	0	-4i	4i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of M₄(2):₄C₄
►On 16 points - transitive group 16T90

Generators in S₁₆

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

(2 6)(4 8)(9 13)(11 15)

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

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

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

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

G:=TransitiveGroup(16,90);

►On 16 points - transitive group 16T110

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,110);

M₄(2):₄C₄ is a maximal subgroup of
C₂³.₅C₄² C₈oD₄:C₄ C₂⁴.₆(C₂xC₄) (C₂xQ₈).₂₁₁D₄ C₈:C₄:₁₇C₄ M₄(2).₄₁D₄ (C₂xD₄).Q₈ M₄(2).₄₄D₄ (C₂xC₈):D₄ M₄(2).₄₆D₄ M₄(2).₄₇D₄ C₄².₅D₄ C₄².₆D₄ C₄².₄₂₆D₄ M₄(2):₂₁D₄ M₄(2).₅₀D₄ C₄².₄₂₇D₄ M₄(2):₅D₄ M₄(2).D₄ C₄².₈D₄ M₄(2).₈D₄ M₄(2).₉D₄ C₄².₉D₄ (C₂xC₈).D₄ (C₂xC₈).₆D₄ C₄².₃₂Q₈ C₂²:C₄.Q₈ C₂²:C₄.F₅ (C₂xC₈):F₅ M₄(2):₄F₅
(C₂xC_4p).Q₈: C₈.(C₄:C₄) M₄(2).₂₇D₄ (C₂xC₁₂).Q₈ (C₂xC₂₄):C₄ M₄(2):₄Dic₃ C₂₀.₆₀(C₄:C₄) (C₂xC₄₀):C₄ M₄(2):₄Dic₅ ...
M₄(2):₄C₄ is a maximal quotient of
C₄².₂Q₈ C₂³.₂₁C₄² C₄².₃Q₈ C₄².₂₃D₄ C₄².₆Q₈ C₄².₂₆D₄ C₄².₉Q₈ C₄².₃₇₀D₄ C₂³.C₄² C₂³.₈C₄² C₄².₃₀D₄ C₄².₃₁D₄ C₂²:C₄.F₅ (C₂xC₈):F₅ M₄(2):₄F₅
(C₂xC₄).D_4p: C₄².₅Q₈ C₄².₃₈₉D₄ C₄².₁₀Q₈ (C₂xC₁₂).Q₈ (C₂xC₂₄):C₄ M₄(2):₄Dic₃ C₂₀.₆₀(C₄:C₄) (C₂xC₄₀):C₄ ...

Matrix representation of M₄(2):₄C₄ ►in GL₄(F₅) generated by

0	3	0	0
0	0	0	2
1	0	0	0
0	0	4	0

0	0	0	2
0	0	4	0
0	4	0	0
3	0	0	0

2	0	0	0
0	0	2	0
0	3	0	0
0	0	0	3

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

M₄(2):₄C₄ in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4C_4

% in TeX

G:=Group("M4(2):4C4");

// GroupNames label

G:=SmallGroup(64,25);

// by ID

G=gap.SmallGroup(64,25);

# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,88]);

// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^5,c*a*c^-1=a*b,c*b*c^-1=a^4*b>;

// generators/relations

Export

Subgroup lattice of M₄(2):₄C₄ in TeX
Character table of M₄(2):₄C₄ in TeX

G = M4(2):4C4 order 64 = 26

4th semidirect product of M4(2) and C4 acting via C4/C2=C2

G = M₄(2):₄C₄ order 64 = 2⁶

4^th semidirect product of M₄(2) and C₄ acting via C₄/C₂=C₂