C₃×#7341 #10625948

order	768
nilpotency class	5
isoclinism	isoclinic class of #7338
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7341

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#1316
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

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

normal subgroups only

This group has 324 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)(4,5,7,12,8,13,20,27,9,11,19,25,22,26,32,34)(6,14,15,29,16,28,30,35,17,10,18,23,21,24,31,33);(4,6,9,17)(5,10,11,14)(7,18,19,15)(8,21,22,16)(12,23,25,29)(13,28,26,24)(20,30,32,31)(27,35,34,33)

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, 329);