C₃×#10936 #10629543

order	768
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#10936

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator	C₂³	C₂×C₃×C₄²
Frattini, cosocle	C₂⁵	C₂³×C₃
prime radical	C₂⁵×C₃	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂⁵×C₃ ◁ G

1258 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 302 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) = 27
transitive degree	μ_t(G) = 192

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

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

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, 3924);