C₃×#15258 #10633865

order	768
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

5134 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 5134 conjugacy classes of subgroups — too many to draw. Only its 494 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 — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8246);