C₃×#9863 #10628470

order	768
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#9863

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

412 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 142 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)(4,5)(6,12)(7,15)(8,10)(9,13)(11,16)(14,18)(17,19)(20,21,24,28,25,29,23,26)(22,30,33,27,31,34,32,35);(4,6,7,13,8,14,16,19)(5,9,11,12,10,17,15,18)(21,26,29,28)(22,31)(27,34,35,30)(32,33);(5,10)(9,17)(11,15)(12,18)(20,22,23,32,25,31,24,33)(21,27,28,34,29,35,26,30)

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, 2851);