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×#62831 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(5,9)(7,14)(10,15)(13,18)(20,21,26,31)(22,32,34,27)(23,30,25,28)(24,29,33,35);(6,12)(10,15)(13,18)(17,19)(20,21)(22,32)(23,28)(24,35)(25,30)(26,31)(27,34)(29,33);(4,5,8,9)(6,10,12,15)(7,11,14,16)(13,17,18,19)(20,22,26,34)(21,27,31,32)(23,33,25,24)(28,35,30,29);(20,23,26,25)(21,28,31,30)(22,33,34,24)(27,35,32,29);(4,6)(5,10)(7,13)(8,12)(9,15)(11,17)(14,18)(16,19)(20,24)(21,29)(22,23)(25,34)(26,33)(27,28)(30,32)(31,35);(4,7)(5,11)(6,13)(8,14)(9,16)(10,17)(12,18)(15,19)(20,25,26,23)(21,30,31,28)(22,24,34,33)(27,29,32,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55819);