C₃²×#2102 #11769151

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#2102

Subgroups

	subgroup	G/H
center	C₃²×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂×C₃²	#400
prime radical	C₂³×C₃²	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ G

474 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3)(7,8,11,10,12,14,16,20)(9,15,17,22,18,19,21,13);(4,5,6)(7,9,16,17,12,18,11,21)(8,13,20,15,14,22,10,19);(7,10,11,14,12,20,16,8)(9,15,17,22,18,19,21,13)

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, 3162);