C₃×#10009 #10628616

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

Aliases

direct product

C₃×#10009

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

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

normal subgroups only

This group has 528 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)(4,5,10,15)(6,16,7,18)(8,14,9,13)(11,19,12,17)(20,21,24,28)(22,26,30,35)(23,25,31,27)(29,34,32,33);(4,6)(5,11)(7,10)(8,19)(9,17)(12,15)(13,18)(14,16)(21,25)(23,31)(26,34)(27,28)(29,32)(33,35);(4,7,10,6)(5,12,15,11)(8,19,9,17)(13,18,14,16)(20,22,24,30)(21,26,28,35)(23,32,31,29)(25,33,27,34)

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, 2997);