Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,19)(15,18)(16,20)(17,22)(21,26)(23,24)(25,27);(5,9)(6,10)(12,13)(14,19)(15,23)(16,20)(17,22)(18,24)(21,25)(26,27);(4,6,8,10)(5,7,9,11)(12,14)(13,17)(15,21)(16,22)(18,25)(19,20)(23,26)(24,27);(4,7)(5,10)(6,9)(8,11)(12,15)(13,18)(14,21)(16,24)(17,25)(19,26)(20,23)(22,27);(12,14)(13,19)(15,21)(16,22)(17,20)(18,26)(23,25)(24,27);(12,13)(14,17)(15,18)(16,20)(19,22)(21,25)(23,24)(26,27)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56037);