C₉×#2213 #11766934

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2213

Subgroups

	subgroup	G/H
center	C₂×C₉	#459
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#459
prime radical	C₃×#2213	C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₄×C₈×C₉ ◁ G

255 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 90 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)(11,14)(12,17)(13,15)(18,19)(20,25)(21,24)(22,23);(10,11,15,18,16,19,13,14)(12,17,23,25,21,24,20,22);(10,12)(11,17)(13,20)(14,22)(15,23)(16,21)(18,25)(19,24)

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, 945);