Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#10675 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13,17,19,16,14,15,18);(4,6,7,11)(5,8,9,10)(12,14,16,13)(15,18,17,19)(20,21)(22,24)(23,26)(25,27);(12,13,16,14)(15,19,17,18)(21,24)(22,27)(25,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3663);