C₃×#7351 #10625958

order	768
nilpotency class	5
isoclinism	isoclinic class of #7349
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7351

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1317
commutator	C₂⁴	C₂×C₃×C₈
Frattini, cosocle	#407	C₂²×C₃
prime radical	C₂⁴×C₃	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#407 ◁ G

178 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 35 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 339);