Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C3×#35317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3406 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)(5,7)(9,10)(12,13,17,20)(14,15,18,23)(16,24,25,19)(21,27,26,22);(8,9,11,10)(12,14)(13,15)(16,21)(17,18)(19,22)(20,23)(24,27)(25,26);(4,5,6,7)(8,10,11,9)(12,15)(13,18)(14,20)(16,22)(17,23)(19,26)(21,24)(25,27);(12,15)(13,18)(14,20)(16,22)(17,23)(19,26)(21,24)(25,27);(12,16,17,25)(13,19,20,24)(14,21,18,26)(15,22,23,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28305);