C₂³×C₃×#120 #10681262

order	768
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃ · Schur multiplier (H₂) = C₂¹⁶

Aliases

C₂³×C₃×#120

	subgroup	G/H
center, socle	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

42862 conjugacy classes of subgroups; only the 7902 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 7902 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 6 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 55643);