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×#62821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7338 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)(5,8)(9,11)(12,13)(14,23)(15,26)(16,17)(18,19)(20,24)(21,25)(22,27);(6,10)(9,11)(13,19)(14,24)(16,22)(21,26);(4,5)(6,9)(7,8)(10,11)(12,14)(13,20)(15,22)(16,25)(17,26)(18,24)(19,23)(21,27);(12,15,18,25)(13,21,19,26)(14,22,24,16)(17,20,27,23);(4,6)(5,9)(7,10)(8,11)(12,16,18,22)(13,17,19,27)(14,25,24,15)(20,26,23,21);(4,5)(6,9)(7,8)(10,11)(12,17)(13,22)(14,26)(15,20)(16,19)(18,27)(21,24)(23,25)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55809);