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C₄²×M₁₆ #20328

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

C₄²×M₁₆

Subgroups

	subgroup	G/H
center	C₄³	C₂²
commutator	C₂	C₂×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	M₁₆×C₄²⋊C₃
maximal subgroup of	#94904, #94905, #94908, …
maximal normal subgroup of	#94904, #94905, #94908, …
maximal quotient of	C₄×C₈×M₁₆, C₄×#1451, #180800, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13316);