C₃²×#1655 #11768704

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

786 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 288

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2715);