C₉×#1664 #11766385

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1664

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₄²	C₂³×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂×C₃×D₄	C₃×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

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

normal subgroups only

This group has 276 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) = 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)(10,11,15,19)(12,20,22,25)(13,18,14,17)(16,23,24,21)(26,27,31,29)(28,34,37,41)(30,38,40,33)(32,39,35,36);(10,12)(11,16)(13,23)(14,21)(15,22)(17,25)(18,20)(19,24)(26,28,31,37)(27,32,29,35)(30,39,40,36)(33,41,38,34);(10,12)(11,16)(13,21)(14,23)(15,22)(17,20)(18,25)(19,24)(26,29,31,27)(28,35,37,32)(30,33,40,38)(34,39,41,36)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 396);