#10714613

order	768
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²	#5032
commutator, prime radical, cosocle	C₂²×C₃×D₄	C₂³
Frattini	C₂²×D₄	C₂²×D₃
Fitting	C₃×#1890	C₂
socle	C₂²×C₃	B(4,𝔽₂)
hypercenter	#1890	D₃
hypocenter	C₃	#13421
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #1890

496 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

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

μ_t(G) = 192

Polycyclic presentation:

2^8,3|a^2=h,b^2=h,c^2=d,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eg,i^b=i^2,g^c=gh,f^e=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 88994);