Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5)(6,7)(8,9)(10,11)(13,17)(15,22)(18,23)(21,26);(5,9)(7,10)(14,20)(18,23)(21,26)(25,27);(4,5,8,9)(6,10,11,7)(12,13,16,17)(14,18,20,23)(15,19,22,24)(21,25,26,27);(4,6)(5,10)(7,9)(8,11);(4,7,8,10)(5,6,9,11)(12,14)(13,18)(15,21)(16,20)(17,23)(19,25)(22,26)(24,27);(4,6)(5,10)(7,9)(8,11)(12,15)(13,19)(14,21)(16,22)(17,24)(18,25)(20,26)(23,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55812);