C₃²×#2198 #11769247

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2198

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₃²
prime radical	C₃²×#453	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃² ◁ C₂×C₃²×D₄ ◁ G

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

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 168 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)(9,17)(10,14)(11,15)(12,16)(13,18)(19,21)(20,22);(4,5,6)(7,9,12,10)(8,13,16,21)(11,18,20,19)(14,15,17,22);(7,10,12,9)(8,14,16,17)(11,18,20,19)(13,15,21,22)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3258);