C₉×#2240 #11766961

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#2240

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₈
prime radical	C₃×#493	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

240 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 33
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 33 points:

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

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, 972);