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×#62822 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7274 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)(12,13)(14,15)(16,24)(17,18)(19,20)(21,25)(22,26)(23,27);(5,8)(9,10)(13,18)(14,20)(21,24)(22,27);(4,5)(6,9)(7,8)(10,11)(12,14,17,20)(13,19,18,15)(16,22,25,27)(21,26,24,23);(4,6,7,11)(5,9,8,10)(12,15,17,19)(13,20,18,14)(16,23,25,26)(21,27,24,22);(12,13)(14,19)(15,20)(16,21)(17,18)(22,26)(23,27)(24,25);(12,16)(13,21)(14,22)(15,23)(17,25)(18,24)(19,26)(20,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55810);