C₃×#19083 #10637690

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

Aliases

direct product

C₃×#19083

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 876 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) = 192

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

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, 12071);