C₉×#2053 #11766774

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2053

Subgroups

	subgroup	G/H
center	C₄×C₉	#124
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#394
prime radical	C₃×#575	C₂×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

396 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 120 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)(12,21)(13,26)(14,31)(15,19)(16,20)(17,23)(18,28)(22,32)(24,34)(25,33)(27,35)(29,36)(30,39)(37,40)(38,41);(10,12)(11,17)(13,27)(14,24)(15,23)(16,25)(18,35)(19,33)(20,34)(21,31)(22,29)(26,32)(28,38)(30,37)(36,40)(39,41);(10,13,14,28,16,30,15,29)(11,18,19,26,20,36,31,39)(12,22,23,37,25,38,24,27)(17,32,21,35,34,41,33,40)

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, 785);