C₃×#13361 #10631968

order	768
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13361

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₃
prime radical	C₃×#3021	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ G

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 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)(4,5,9,14)(6,15,19,30)(7,20,24,12)(8,11,25,13)(10,21,28,17)(16,33,18,32)(22,31,23,29)(26,34,27,35);(4,6)(5,10)(7,21)(8,16)(9,19)(11,27)(12,30)(13,26)(14,28)(15,20)(17,24)(18,25)(22,35)(23,34)(29,32)(31,33);(5,11)(6,16)(8,25)(12,29)(13,14)(15,30)(17,35)(18,19)(20,31)(21,34)(22,23)(26,27)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6349);