C₉×#1287 #11766008

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₉×#1287

Subgroups

	subgroup	G/H
center	C₂²×C₉	#95
commutator	C₂×C₄	C₄²×C₉
Frattini, cosocle	C₂×C₃×C₄²	C₂²×C₃
socle	C₂²×C₃	C₃×#95
prime radical	C₂³×C₃	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

189 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 576

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

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

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, 19);