C₃×#19080 #10637687

order	768
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#19080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₃×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

936 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 35 points:

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

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, 12068);