C₉×#1868 #11766589

order	1152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#1868

Subgroups

	subgroup	G/H
center	C₄×C₉	#120
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#385
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

390 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 288

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

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 600);