C₃²×#555 #10596356

order	576
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂³×C₃×C₄

Aliases

direct product

C₃²×#555

Subgroups

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

708 conjugacy classes of subgroups; only the 450 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 450 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)(10,14)(11,12)(16,17)(20,21);(4,5,6)(8,12)(10,13)(15,16,19,17)(18,21,22,20);(7,9,11,14)(8,13,12,10)(15,17,19,16)(18,21,22,20);(7,10,11,13)(8,14,12,9)(15,18,19,22)(16,20,17,21)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11779076, C₃×#71927, #11779486, …
maximal subgroup of	C₃²×#1624, C₃²×#1634, C₃²×#1646, …
maximal normal subgroup of	C₃²×#1624, C₃²×#1634, C₃²×#1646, …
maximal quotient of	C₃²×#2346, C₃²×#2354, C₃²×#2363, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8238);