C₃×#7760 #10626367

order	768
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#7760

Subgroups

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

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

normal subgroups only

This group has 424 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) = 768

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

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

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, 748);