C₉×#2381 #11767102

order	1152
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₉×#2381

Subgroups

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, prime radical, cosocle	C₂³×C₃	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

636 conjugacy classes of subgroups; only the 396 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 396 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 1113);