C₃×#14678 #10633285

order	768
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃×#14678

Subgroups

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

1182 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1182 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) = 768

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

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7666);