C₉×#497 #10589051

order	576
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#497

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₄
prime radical	C₂×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

186 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 29
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11772812, #11780556, #11780588, …
maximal subgroup of	C₉×#1407, C₉×#1659, C₉×#1660, …
maximal normal subgroup of	C₉×#1407, C₉×#1659, C₉×#1660, …
maximal quotient of	C₉×#1714, C₉×#1716, C₉×#1718, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 933);