Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(17,23)(24,27);(4,5,7,9)(6,11,10,8)(12,13)(14,17)(15,18)(16,19)(20,24)(21,23)(22,25)(26,27);(12,14)(13,17)(15,20)(16,21)(18,24)(19,23)(22,26)(25,27);(12,15,16,22)(13,18,19,25)(14,20,21,26)(17,24,23,27)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11112);