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×#62924 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7380 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)(6,10)(7,8)(9,11)(12,13)(14,21)(15,25)(16,19)(17,26)(18,20)(22,27)(23,24);(5,8)(9,10)(12,14,18,16)(13,19,20,21)(15,23,26,22)(17,24,25,27);(4,5)(6,9)(7,8)(10,11)(12,15)(13,17)(14,22)(16,23)(18,26)(19,27)(20,25)(21,24);(4,6)(5,9)(7,11)(8,10)(12,15)(13,17)(14,23)(16,22)(18,26)(19,24)(20,25)(21,27);(12,16,18,14)(13,19,20,21)(15,23,26,22)(17,27,25,24);(12,17)(13,15)(14,24)(16,27)(18,25)(19,23)(20,26)(21,22)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55912);