C₃×#13373 #10631980

order	768
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13373

Subgroups

	subgroup	G/H
center	C₃×C₄	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#2023
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,15,17,32)(7,20)(8,13)(9,25)(10,14)(11,18,27,34)(12,24)(16,33,19,29)(21,31)(22,30,35,26)(23,28);(6,16,17,19)(7,21)(11,26,27,30)(12,28)(15,29,32,33)(18,35,34,22)(20,31)(23,24);(4,6)(5,11)(7,22)(8,19)(9,16)(10,17)(12,29)(13,30)(14,27)(15,31)(18,23)(20,32)(21,35)(24,34)(25,26)(28,33)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72271432, #72271439, D₃×#13373, …
maximal subgroup of	C₃×#500001, #72271432, #72271439, …
maximal normal subgroup of	C₃×#500001, #72271432, #72271439, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6361);