C₃×#29658 #10648265

order	768
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₃×#29658

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

1086 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 22646);