Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
2186 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)(6,10)(7,9)(8,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27);(4,6,7,11)(5,8,9,10);(6,11)(8,10)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(5,9)(8,10)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21);(13,20)(18,27)(19,24)(21,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53789);