C₉×#1818 #11766539

order	1152
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#1818

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₃×C₄
prime radical	C₂⁴×C₃	C₂×C₃×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

438 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 576

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)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(15,18)(19,21)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(11,13)(14,16,17,20)(15,19,18,21)(24,30)(27,33)(31,36)(34,37)

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 550);