C₃×#9202 #10627809

order	768
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9202

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

516 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 192

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2190);