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Q₃₂×#272 #13144221

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	H(3,𝔽₃)×Q₃₂ ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ C₂×Q₃₂

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	D₈×C₃⋊D₃
commutator, cosocle	C₈×H(3,𝔽₃)	C₂³
Frattini	C₃×C₈	C₂²×C₃⋊D₃
Fitting	H(3,𝔽₃)×Q₃₂	C₂
prime radical	C₂×#272	D₈
hypercenter	C₃×Q₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×Q₃₂
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×Q₃₂

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

normal subgroups only

This group has 451 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.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 288

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10284);