C₃×#13545 #10632152

order	768
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#13545

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2101
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#405 ◁ G

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

normal subgroups only

This group has 216 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)(4,5,8,13,9,11,21,28)(6,14,24,27,16,10,18,31)(7,19,22,35,23,12,20,30)(15,29,17,26,33,25,34,32);(4,6)(5,10,11,14)(7,15)(8,24)(9,16)(12,29,19,25)(13,31,28,27)(17,22)(18,21)(20,34)(23,33)(26,35,32,30);(5,11)(6,15,16,33)(7,20)(8,21)(10,25,14,29)(12,30)(17,24,34,18)(19,35)(22,23)(26,27,32,31)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6533);