C₃×#9517 #10628124

order	768
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9517

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

520 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 35 points:

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

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, 2505);