C₃×#10508 #10629115

order	768
nilpotency class	3
isoclinism	isoclinic class of #10508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#10508

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3213	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

414 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3496);