C₉×#2050 #11766771

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2050

Subgroups

	subgroup	G/H
center	C₄×C₉	#124
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#394
prime radical	C₃×#575	C₂×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

399 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,19)(13,21)(14,22)(15,18)(16,20)(17,24)(23,25);(10,12)(11,16)(13,17)(14,23)(15,21)(18,25)(19,22)(20,24);(10,13,14,16,15,19,24,25)(11,12,17,20,18,21,22,23)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 782);