Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,18)(16,19)(20,23)(21,25)(22,24)(26,27);(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,22)(16,21)(18,24)(19,25)(20,26)(23,27);(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(14,21)(17,25)(22,27)(24,26);(13,19)(17,25)(18,20)(24,26)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53478);