C₉×#1324 #11766045

order	1152
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1324

Subgroups

	subgroup	G/H
center	C₂²×C₉	#99
commutator	C₂×C₄	C₂×C₈×C₉
Frattini, cosocle	C₂×C₃×C₄⋊C₄	C₂²×C₃
socle	C₂²×C₃	C₃×#99
prime radical	C₂³×C₃	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄ ◁ G

177 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 288

A minimal faithful permutation representation — 2 generators on 33 points:

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

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, 56);