C₃×#10010 #10628617

order	768
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10010

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

534 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 102 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)(4,5,9,13)(6,14,16,19)(7,12,8,11)(10,17,18,15)(20,21,24,27)(22,28,30,35)(23,31,33,26)(25,32,34,29);(4,6,9,16)(5,10,13,18)(7,17,8,15)(11,19,12,14)(20,22,24,30)(21,25,27,34)(23,32,33,29)(26,35,31,28);(4,6)(5,10)(7,15)(8,17)(9,16)(11,14)(12,19)(13,18)(23,33)(25,34)(26,31)(29,32)

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, 2998);