C₃×#8745 #10627352

order	768
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#8745

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#381
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

384 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,13,15,9)(7,12)(8,11)(10,14,17,19)(16,18)(20,21,24,27,23,26,31,34)(22,28,30,35,29,25,32,33);(4,6)(5,9)(7,14)(8,15)(10,12)(11,13)(16,19)(17,18)(20,22,23,29)(21,25,26,28)(24,30,31,32)(27,33,34,35);(4,6)(5,10,11,17)(7,14)(8,15)(9,12,13,18)(16,19)(21,26)(24,31)(25,28)(30,32)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1733);