C₃×#20772 #10639379

order	768
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#20772

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

1062 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 462 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) = 384

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13760);