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×#62944 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7214 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)(6,10)(7,8)(9,11)(13,18)(14,20)(16,24)(23,26);(5,8)(9,10)(12,13)(14,19)(15,20)(16,21)(17,18)(22,26)(23,27)(24,25);(4,5)(6,9)(7,8)(10,11)(12,14)(13,15)(16,22)(17,20)(18,19)(21,23)(24,27)(25,26);(4,6,7,11)(5,9,8,10)(12,14)(13,19)(15,18)(16,22)(17,20)(21,26)(23,25)(24,27);(12,15,17,19)(13,20,18,14)(16,23,24,26)(21,27,25,22);(12,16)(13,21)(14,22)(15,23)(17,24)(18,25)(19,26)(20,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55932);