C₃×#7303 #10625910

order	768
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7303

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#340
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
prime radical	C₂³×C₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

204 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 43
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3)(4,5,8,10,9,6,7,11)(12,13,16,22,19,24,18,34)(14,25,27,40,29,41,32,36)(15,23,17,33,31,39,30,21)(20,28,35,26,38,42,37,43);(4,6,9,5)(7,10,8,11)(12,14,17,28,16,32,31,43,19,29,30,42,18,27,15,26)(13,20,23,36,34,35,21,25,24,38,39,40,22,37,33,41)

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, 291);