C₃×#7364 #10625971

order	768
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7364

Subgroups

	subgroup	G/H
center	C₃×C₈	#99
commutator	C₂×C₄	C₂×C₃×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂²×C₃
socle	C₂×C₃	#1322
prime radical	C₂³×C₃	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₂×C₃×M₃₂ ◁ G

126 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 54 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)(4,5,7,13,8,15,20,28,9,12,19,25,21,26,32,34,10,14,18,27,22,29,31,35,6,11,16,23,17,24,30,33);(5,11,14,12)(7,18)(13,23,27,25)(15,24,29,26)(16,19)(20,31)(28,33,35,34)(30,32)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 352);