Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×#41 ◁ G | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(14,20)(15,22)(17,24)(19,26)(21,23)(25,27);(5,8)(9,11)(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);(4,6,7,10)(5,9,8,11);(13,18)(17,25)(19,26)(24,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53560);