C₃×#9839 #10628446

order	768
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₃×#9839

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

542 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 214 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)(4,5,6,7)(9,12)(10,13)(14,15)(16,17)(18,24)(19,21)(20,28)(22,26)(23,27)(25,30)(29,31);(5,7)(8,9,13,15,11,14,10,12)(17,21)(18,25)(23,28)(26,31);(16,18,20,26,19,25,27,31)(17,22,23,24,21,29,28,30)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2827);