C₃×#33567 #10652174

order	768
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#33567

Subgroups

	subgroup	G/H
center	C₃×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₃
socle	C₂×C₃	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ G

1184 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3)(4,5)(6,14)(7,12)(8,11)(9,18)(10,15)(13,16)(17,19);(5,11)(6,9)(8,16)(12,14)(13,15)(18,19);(4,6)(7,16)(8,17)(9,10)(11,13)(14,18);(4,7,10,17)(5,12,15,19)(6,16,9,8)(11,14,13,18)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26555);