C₃×#12331 #10630938

order	768
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12331

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₃×#2206	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×Q₈ ◁ G

328 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,8)(6,10)(9,11)(13,18)(14,17)(15,22)(19,27)(20,26)(21,24)(23,25);(6,10)(8,11)(12,13,16,20)(14,19,17,25)(15,23,24,27)(18,21,26,22);(4,5,7,9)(6,11,10,8)(12,14,16,17)(13,19,20,25)(15,21,24,22)(18,23,26,27)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5319);