C₉×#2193 #11766914

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2193

Subgroups

	subgroup	G/H
center	C₂×C₉	#447
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#447
prime radical	C₃×#2193	C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131 ◁ G

315 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9)(11,17)(12,21)(13,24)(14,15)(18,32)(19,36)(20,31)(22,38)(23,28)(25,29)(26,27)(30,33)(34,41)(35,39);(10,11)(12,18)(13,25)(14,19)(15,29)(16,20)(17,24)(21,33)(22,39)(23,35)(26,31)(27,36)(28,34)(30,37)(32,38)(40,41);(10,12,16,23)(11,18,20,35)(13,22,27,38)(14,28,15,21)(17,30,31,41)(19,34,29,33)(24,37,26,40)(25,39,36,32)

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, 925);