C₃×#7297 #10625904

order	768
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7297

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#102
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
socle	C₂²×C₃	#340
prime radical	C₂³×C₃	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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)(12,13,16,22,19,24,18,35)(14,25,27,37,30,39,29,43)(15,31,33,40,34,23,17,21)(20,28,36,32,38,42,41,26);(4,6,9,5)(7,10,8,11)(12,14,17,28,18,29,34,26,19,30,33,42,16,27,15,32)(13,20,23,37,22,36,21,39,24,38,31,43,35,41,40,25)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 285);