C₃×#8802 #10627409

order	768
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#8802

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#117
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#383
prime radical	C₂³×C₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

258 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 110 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)(6,14,17,18)(7,10)(8,12)(9,13)(11,16,19,15)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(5,10,13,12)(11,18,19,14)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(4,6,7,15,9,17,8,16)(5,11,12,14,13,19,10,18)(22,28)(25,31)(29,34)(32,35)

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, 1790);