C₃×#15172 #10633779

order	768
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₃×#15172

Subgroups

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

2826 conjugacy classes of subgroups; only the 478 normal subgroups are shown.

normal subgroups only

This group has 2826 conjugacy classes of subgroups — too many to draw. Only its 478 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8160);