C₃×#33089 #10651696

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

Aliases

direct product

C₃×#33089

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

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

normal subgroups only

This group has 728 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)(22,27)(23,28)(24,33)(25,34)(26,30)(29,31)(32,35);(4,6,8,7)(5,9,10,11)(12,14,15,18)(13,16,17,19)(20,22,26,25)(21,27,30,34)(23,32,24,31)(28,35,33,29);(4,7,8,6)(5,9,10,11)(13,17)(16,19)(20,23,26,24)(21,28,30,33)(22,31,25,32)(27,29,34,35);(20,24,26,23)(21,29,30,35)(22,31,25,32)(27,28,34,33)

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, 26077);