C₃²×#2200 #11769249

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

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

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

normal subgroups only

This group has 780 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.

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3260);