C₃×#32933 #10651540

order	768
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₃×#32933

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×#41 ◁ G

1494 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 310 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)(6,7)(8,9)(10,11)(13,17)(15,18)(19,23)(20,24)(22,25)(26,27);(12,13,18,24,16,20,15,17)(14,19,25,27,21,26,22,23);(5,7)(12,14)(13,19)(15,22)(16,21)(17,23)(18,25)(20,26)(24,27);(9,11)(14,21)(19,26)(22,25)(23,27)

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, 25921);