C₃×#12417 #10631024

order	768
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₃×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

240 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(5,8)(6,10)(9,11)(13,17)(14,20)(15,22)(18,27)(19,26)(21,24)(23,25);(4,5,6,8,7,9,10,11);(12,13,16,19)(14,21,23,27)(15,24,22,18)(17,25,26,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5405);