C₃×#10228 #10628835

order	768
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

828 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(8,9)(10,12)(11,14)(13,15)(16,17)(18,20)(19,22)(21,23);(4,5)(6,7)(9,12)(10,15)(13,14);(5,7)(17,20)(18,23)(21,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3216);