C₃×#13327 #10631934

order	768
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13327

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂²×C₃×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

594 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 27 points:

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

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, 6315);