C₉×#1321 #11766042

order	1152
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1321

Subgroups

	subgroup	G/H
center	C₄×C₉	#99
commutator	C₂×C₄	C₂×C₈×C₉
Frattini, cosocle	C₂×C₃×M₁₆	C₂²×C₃
socle	C₂×C₃	C₃×#323
prime radical	C₂×C₃×#41	C₃×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×M₁₆ ◁ G

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

normal subgroups only

This group has 168 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)(10,11,14,19,15,18,29,35,16,20,30,37,13,25,27,36)(12,21,23,38,22,17,31,32,24,34,28,40,26,33,39,41);(10,12)(11,17)(13,26)(14,28)(15,22)(16,24)(18,34)(19,32)(20,33)(21,25)(23,30)(27,31)(29,39)(35,40)(36,38)(37,41)

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, 53);