C₉×#2026 #11766747

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

Aliases

direct product

C₉×#2026

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

387 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 150 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)(14,15,17,19)(16,20,21,18)(22,23,25,27)(24,30,31,37)(26,34,33,29)(28,35,36,32);(10,11)(12,13)(14,16,17,21)(15,18,19,20)(23,27)(28,36)(29,34)(30,37);(11,13)(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30)

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, 758);