M₁₆×#442 #72279518

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

M₁₆×#442

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	C₂³×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2,4,3)(13,14,17,20)(15,18,19,16);(13,15,18,20,17,19,16,14);(1,2,4,3)(5,6,9,12,8,10,7,11)(14,19)(15,20)(16,18);(6,10)(11,12);(1,3,4,2)(6,10)(11,12)(13,16,17,18)(14,15,20,19)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#442, M₁₆×AGL(1,ℤ/16), #72342656
maximal normal subgroup of	AGL(1,ℤ/8)×#442, M₁₆×AGL(1,ℤ/16), #72342656
maximal quotient of	M₁₆×AGL(1,ℤ/16)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,3)(13,14,17,20)(15,18,19,16), (13,15,18,20,17,19,16,14), (1,2,4,3)(5,6,9,12,8,10,7,11)(14,19)(15,20)(16,18), (6,10)(11,12), (1,3,4,2)(6,10)(11,12)(13,16,17,18)(14,15,20,19));

M16×#442 #72279518