C₃×#13357 #10631964

order	768
nilpotency class	4
isoclinism	isoclinic class of #13357
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13357

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂³×C₃
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈ ◁ G

772 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 96

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

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#13357, #72255492
maximal subgroup of	C₃×#498909, D₃×#13357, #72255492
maximal normal subgroup of	C₃×#498909, D₃×#13357, #72255492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6345);