C₃×#13459 #10632066

order	768
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13459

Subgroups

	subgroup	G/H
center	C₃×C₄	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#2058
prime radical	C₂³×C₃	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

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

normal subgroups only

This group has 318 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)(6,15,17,32)(7,20)(8,13)(9,25)(10,14)(11,18,27,34)(12,24)(16,29,19,31)(21,35)(22,26,33,30)(23,28);(6,16,17,19)(7,21)(11,26,27,30)(12,28)(15,31,32,29)(18,22,34,33)(20,35)(23,24);(4,6,9,16,10,17,8,19)(5,11,13,30,14,27,25,26)(7,22,24,18,21,33,23,34)(12,29,20,15,28,31,35,32)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#13459, #72270355, #72271588, …
maximal subgroup of	C₃×#500349, D₃×#13459, #72270355, …
maximal normal subgroup of	C₃×#500349, D₃×#13459, #72270355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6447);