Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#38321 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(13,17)(18,19)(20,21,23,24)(22,26,27,25);(5,8)(6,10)(21,24)(25,26);(4,5,7,8)(6,9,10,11)(12,13)(14,18)(15,19)(16,17)(20,21)(22,25)(23,24)(26,27);(12,14,16,15)(13,18,17,19)(20,22)(21,25)(23,27)(24,26);(4,6)(5,9)(7,10)(8,11)(12,15,16,14)(13,19,17,18)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 31309);