C₃×#13340 #10631947

order	768
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13340

Subgroups

	subgroup	G/H
center	C₃×C₄	#493
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#2011
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

552 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5)(6,17)(7,14)(8,13)(9,27)(10,15)(11,18)(12,19)(16,23)(20,29)(21,35)(22,32)(24,34)(25,30)(26,33)(28,31);(4,6)(5,11)(7,22)(8,18)(9,28)(10,21)(12,27)(13,31)(14,17)(15,30)(16,32)(19,24)(20,23)(25,26)(29,33)(34,35);(4,6)(5,12)(7,19)(8,25)(9,28)(10,21)(11,16)(13,17)(14,29)(15,32)(18,26)(20,23)(22,24)(27,30)(31,34)(33,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72270154, #72270159, #72271696
maximal subgroup of	C₃×#499752, #72270154, #72270159, …
maximal normal subgroup of	C₃×#499752, #72270154, #72270159, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6328);