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SL(2,𝔽₃)×#611 #13147172

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

SL(2,𝔽₃)×#611

Splittings

C₂²×Q₈×C₉ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×A₄×D₉
commutator, cosocle	C₂×Q₈×C₉	C₂²×C₃
Frattini, socle	C₂²×C₃	C₂×D₃×A₄
Fitting	C₂²×Q₈×C₉	C₂×C₃
hypercenter	C₂³	A₄×D₉
hypocenter	Q₈×C₉	C₃×D₄
LCS	G ▷ C₂×Q₈×C₉ ▷ Q₈×C₉
UCS	C₁ ◁ C₂² ◁ C₂³

357 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 59 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

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 13235);