C₃×#10237 #10628844

order	768
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10237

Subgroups

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

560 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 31
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3225);