Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,9)(10,11)(14,20)(15,22)(17,24)(19,26)(21,23)(25,27);(4,5,9,11,7,10,6,8)(12,13)(14,17)(15,19)(16,18)(20,24)(21,25)(22,26)(23,27);(12,14,16,21)(13,17,18,25)(15,23,22,20)(19,27,26,24);(13,18)(17,25)(19,26)(24,27)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12386);