Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60563 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| prime radical | C3×#3006 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C42 ◁ G | |
2180 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)(12,13)(14,18)(15,23)(16,20)(17,21)(19,22)(24,26)(25,27);(5,8)(9,11)(12,14,17,16)(13,18,21,20)(15,22,24,25)(19,26,27,23);(12,15)(13,19)(14,22)(16,25)(17,24)(18,26)(20,23)(21,27);(4,6,7,10)(5,9,8,11);(12,14,17,16)(13,20,21,18)(15,22,24,25)(19,23,27,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53551);