C₃×#8139 #10626746

order	768
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#8139

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#376
prime radical	C₂³×C₃	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 192

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1127);