C₃×#7799 #10626406

order	768
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7799

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 404 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) = 384

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(5,9)(6,7)(10,11)(12,13)(14,16)(15,18)(17,19)(20,21,24,27)(22,25,29,32)(23,26,30,33)(28,34,35,31);(4,5)(6,9)(7,11)(8,10)(20,22,24,29)(21,25,27,32)(23,28,30,35)(26,31,33,34);(4,6,8,7)(5,9,10,11)(13,16)(14,19)(17,18)(20,23,24,30)(21,26,27,33)(22,28,29,35)(25,31,32,34)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 787);