C₃×#10022 #10628629

order	768
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10022

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#2631	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

576 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,5,8,15)(6,10,17,12)(7,11,20,29)(9,16,22,14)(13,23,30,21)(18,27,31,26)(19,34,32,28)(24,35,25,33);(5,10)(6,17)(7,20)(11,28)(12,15)(13,24)(14,26)(16,27)(18,31)(21,23)(25,30)(29,34);(5,11)(6,18)(7,21)(10,24)(12,25)(13,14)(15,29)(16,30)(17,31)(19,32)(20,23)(26,28)(27,34)(33,35)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#246930
maximal normal subgroup of	C₃×#246930

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3010);