C₃²×#540 #10596341

order	576
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product

C₃²×#540

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

720 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2,3)(7,8)(9,13)(10,11)(12,14)(16,19)(17,21);(4,5,6)(8,11)(12,13)(15,16,18,19)(17,20,21,22);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22);(7,9,10,14)(8,12,11,13)(15,17,18,21)(16,20,19,22)

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11872237, C₃×#83434, #11872699, …
maximal subgroup of	C₃²×#3119, C₃²×#3122, C₃²×#3126, …
maximal normal subgroup of	C₃²×#3119, C₃²×#3122, C₃²×#3126, …
maximal quotient of	C₃²×#2314, C₃²×#2319, C₃²×#2324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8223);