C₃×#13457 #10632064

order	768
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 356 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) = 23
transitive degree	μ_t(G) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6445);