#11794313

order	1152
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²	#64241
commutator, prime radical, cosocle	C₂³×C₃⋊D₃	C₂³
Frattini	C₂³	C₂×GO⁺(4,𝔽₂)
Fitting	C₂²×C₃²×C₄	D₄
socle	C₂²×C₃²	#120
hypercenter	C₂²×C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	#2062
LCS	G ▷ C₂³×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

552 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

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

μ_t(G) = 96

Polycyclic presentation:

2^7,3^2|a^2=d,b^2=d,c^2=d,b^a=bd,c^a=ce,f^a=fg,c^b=cf,e^b=eg,i^b=hi^2,h^c=hi,i^c=i^2,h^f=h^2,i^f=i^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 28324);