C₃×#33092 #10651699

order	768
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#33092

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

770 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 384

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26080);