C₉×#1391 #11766112

order	1152
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#1391

Subgroups

	subgroup	G/H
center	C₄×C₉	#99
commutator	C₂³	C₄²×C₉
Frattini, prime radical, cosocle	C₂×C₃×#41	C₂²×C₃
socle	C₂×C₃	C₃×#342
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×#41 ◁ G

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

normal subgroups only

This group has 249 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) = 288

A minimal faithful permutation representation — 2 generators on 41 points:

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

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, 123);