C₃×#8526 #10627133

order	768
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8526

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂×C₃×#540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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)(4,5,8,13,9,12,7,11)(6,14,15,19,16,10,17,18)(20,21)(22,29,31,26)(23,27)(24,28)(25,30,34,32)(33,35);(4,6,9,16)(5,10,12,14)(7,15,8,17)(11,18,13,19)(20,22)(21,25)(23,32)(24,31)(26,27)(28,34)(29,35)(30,33);(4,7,9,8)(5,11,12,13)(6,15,16,17)(10,18,14,19)(20,22)(21,26)(23,30)(24,31)(25,27)(28,29)(32,33)(34,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1514);