C₃×#12524 #10631131

order	768
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#12524

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

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

normal subgroups only

This group has 246 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) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5512);