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×#62900 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7428 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)(13,18)(14,20)(16,24)(23,26);(5,8)(9,10)(12,13,17,18)(14,15,20,19)(16,21,24,25)(22,23,27,26);(4,5)(6,9)(7,8)(10,11)(12,14)(13,19)(15,18)(16,22)(17,20)(21,26)(23,25)(24,27);(4,6)(5,9)(7,11)(8,10)(12,14)(13,15)(16,22)(17,20)(18,19)(21,23)(24,27)(25,26);(12,15)(13,20)(14,18)(16,23)(17,19)(21,27)(22,25)(24,26);(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, 55888);