C₉×#2211 #11766932

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2211

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₈
prime radical	C₂×C₃×D₈	C₂²×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

237 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9)(11,13)(14,15)(16,24)(17,20)(18,30)(19,33)(21,26)(22,32)(23,29)(25,40)(27,44)(28,41)(31,39)(34,38)(35,42)(36,43)(37,45);(10,11,12,13)(15,20)(21,35)(22,36)(23,33)(24,40)(30,41)(37,39)(38,44);(14,16,17,25)(15,21,20,35)(18,31,28,45)(19,34,29,27)(22,38,36,44)(23,39,33,37)(24,41,40,30)(26,43,42,32)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 943);