C₃×#13410 #10632017

order	768
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13410

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₃×#3150	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

462 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

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, 6398);