C₉×#2074 #11766795

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2074

Subgroups

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

333 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 37
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 806);