C₃×#33728 #10652335

order	768
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#33728

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₈
prime radical	C₂³×C₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

482 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 270 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 — 4 generators on 43 points:

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

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, 26716);