C₃×#63040 #10681647

order	768
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₃×#63040

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

8508 conjugacy classes of subgroups; only the 5718 normal subgroups are shown.

normal subgroups only

This group has 8508 conjugacy classes of subgroups — too many to draw. Only its 5718 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) = 192

A minimal faithful permutation representation — 6 generators on 19 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 56028);