C₉×#464 #10589018

order	576
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#464

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

168 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 81 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)(16,22)(17,19)(18,26)(20,24)(21,25)(23,29)(27,28);(10,11,12,13)(15,19)(20,28)(21,26)(22,29);(14,16,17,23)(15,20,19,28)(18,27,25,24)(21,29,26,22)

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₉×#464, #11865743, #11865907, …
maximal subgroup of	C₂×C₉×#464, C₉×#3035, C₉×#3037, …
maximal normal subgroup of	C₂×C₉×#464, C₉×#3035, C₉×#3037, …
maximal quotient of	C₉×#1678, C₉×#1680, C₉×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 900);