C₃×#10254 #10628861

order	768
nilpotency class	3
isoclinism	isoclinic class of #10254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

634 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3242);