C₃²×#541 #10596342

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×#541

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
prime radical	C₂³×C₃²	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

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

normal subgroups only

This group has 576 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) = 30
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2,3)(9,13)(11,14)(15,16,19,22)(17,20,25,23)(18,26,27,21)(24,29,30,28);(4,5,6)(7,8,10,12)(9,11,13,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,27)(28,30);(7,9)(8,11)(10,13)(12,14)(15,17)(16,20)(18,24)(19,25)(21,28)(22,23)(26,29)(27,30);(15,18,19,27)(16,21,22,26)(17,24,25,30)(20,28,23,29)

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11872391, C₃×#83446, #11872785, …
maximal subgroup of	C₃²×#3120, C₃²×#3123, C₃²×#3127, …
maximal normal subgroup of	C₃²×#3120, C₃²×#3123, C₃²×#3127, …
maximal quotient of	C₃²×#2314, C₃²×#2320, C₃²×#2323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8224);