C₃²×#1828 #11768877

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1828

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#381
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂⁴	C₂³×C₃²
prime radical	C₂²×C₃²×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂⁴×C₃² ◁ G

888 conjugacy classes of subgroups; only the 300 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 300 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)(9,15,17,12)(10,18)(11,14)(13,19)(16,20,22,21);(4,5,6)(7,9)(8,12)(10,16)(11,17)(13,20)(14,15)(18,21)(19,22);(7,10)(8,13,14,18)(9,16)(11,19)(12,20,15,21)(17,22)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2888);