C₉×#1537 #11766258

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1537

Subgroups

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

291 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 41
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9)(10,11,13,17,14,16,22,24)(12,18,19,25,20,15,21,23)(26,27)(28,35,37,40)(29,38)(30,33)(31,34)(32,36,41,39);(10,12,14,20)(11,15,16,18)(13,21,22,19)(17,25,24,23)(26,28)(27,32)(29,36)(30,39)(31,37)(33,40)(34,41)(35,38);(26,29,31,30)(27,33,34,38)(28,36,37,39)(32,40,41,35)

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, 269);