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×#62903 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5,8,13)(6,14,15,10)(7,18,20,12)(9,11,21,26)(16,23,28,27)(17,22,29,24)(19,31,32,25)(30,34,35,33);(4,6,9,17)(5,10,11,24)(7,16,19,30)(8,15,21,29)(12,23,25,34)(13,14,26,22)(18,27,31,33)(20,28,32,35);(5,11)(6,15)(10,22)(12,25)(13,26)(14,24)(16,28)(17,29)(18,31)(23,33)(27,34)(30,35);(4,7,8,20)(5,12,13,18)(6,16,15,28)(9,19,21,32)(10,23,14,27)(11,25,26,31)(17,30,29,35)(22,33,24,34);(7,19)(12,25)(16,30)(18,31)(20,32)(23,34)(27,33)(28,35);(5,11)(10,24)(12,25)(13,26)(14,22)(18,31)(23,34)(27,33)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55891);