C₉×#2032 #11766753

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2032

Subgroups

	subgroup	G/H
center	C₂×C₉	#393
commutator	C₂⁴	C₂³×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂×C₃	C₃×#393
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂⁴×C₉ ◁ G

417 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 764);