Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62901 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7106 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)(14,21)(18,25)(22,26)(23,27);(5,9)(6,10)(12,13,17,20)(14,18,21,25)(15,16,19,24)(22,23,26,27);(4,6,8,10)(5,7,9,11)(12,14)(13,18)(15,22)(16,23)(17,21)(19,26)(20,25)(24,27);(4,7)(5,10)(6,9)(8,11)(12,15)(13,16)(14,22)(17,19)(18,23)(20,24)(21,26)(25,27);(12,16)(13,19)(14,23)(15,20)(17,24)(18,26)(21,27)(22,25);(12,14)(13,18)(15,22)(16,23)(17,21)(19,26)(20,25)(24,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55889);