C₉×#1385 #11766106

order	1152
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1385

Subgroups

	subgroup	G/H
center	C₄×C₉	#106
commutator	C₈	C₄²×C₉
Frattini, cosocle	C₂²×C₃×C₈	C₂²×C₃
socle	C₂×C₃	C₃×#340
prime radical	C₂³×C₃	C₃×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

186 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 114 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)(10,11,14,20,17,22,16,32)(12,23,25,38,27,39,30,34)(13,21,15,31,29,37,28,19)(18,26,33,24,36,40,35,41);(10,12,15,26,14,30,29,41,17,27,28,40,16,25,13,24)(11,18,21,34,32,33,19,23,22,36,37,38,20,35,31,39)

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, 117);