C₉×#1328 #11766049

order	1152
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1328

Subgroups

	subgroup	G/H
center	C₄×C₉	#99
commutator	C₂³	C₂×C₈×C₉
Frattini, cosocle	C₂×C₃×M₁₆	C₂²×C₃
socle	C₂×C₃	C₃×#324
prime radical	C₂³×C₃	C₂×C₃×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×M₁₆ ◁ G

135 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 60 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)(10,11,13,18,14,19,24,32,15,17,23,29,25,30,36,39)(12,16,21,26,22,28,35,37,20,27,33,31,34,38,41,40);(10,12)(11,16,17,27)(13,21)(14,22)(15,20)(18,31,29,26)(19,28,30,38)(23,33)(24,35)(25,34)(32,40,39,37)(36,41)

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, 60);