C₉×#1543 #11766264

order	1152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#1543

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

369 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 144 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)(18,19)(20,27)(21,25)(22,30)(23,26)(24,28)(29,32)(31,33);(11,14)(12,16)(18,20)(19,24)(21,28)(22,31)(23,29)(25,32)(26,33)(27,30);(18,21,23,22)(19,25,26,30)(20,28,29,31)(24,32,33,27)

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, 275);