(C₂×S₄)⋊#157 #13180193

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×#157
commutator	C₂×C₃²×A₄	C₂×C₄
Fitting, socle	C₂³×C₃²	C₄×D₃
prime radical	C₂×C₃⋊D₃×S₄	C₂
cosocle	C₂×A₄×C₃⋊D₃	C₂²
hypocenter	C₃²×A₄	C₂²⋊C₄
LCS	G ▷ C₂×C₃²×A₄ ▷ C₃²×A₄
UCS	C₁ ◁ C₂

588 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 36

Polycyclic presentation:

2^4,3^3,2^2|a^2=c,b^a=bd,f^a=fg,g^a=fg^2,e^b=e^2,h^b=hi,f^c=f^2,g^c=g^2,h^e=i,i^e=hi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46256);