C₃×#7339 #10625946

order	768
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#7339

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1316
commutator	C₂²×C₄	C₂×C₃×C₈
Frattini, cosocle	#381	C₂²×C₃
prime radical	C₂²×C₃×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₂⁴×C₃ ◁ C₃×#381 ◁ G

356 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 35 points:

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

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, 327);