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×#62948 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6586 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)(4,5)(6,15)(7,13)(8,19)(9,17)(10,14)(11,18)(12,16)(21,25)(26,32)(27,33)(34,35);(4,6)(5,9)(7,8)(10,17)(11,19)(12,13)(14,15)(16,18);(4,7)(5,11)(6,16)(8,17)(9,12)(10,18)(13,14)(15,19)(20,21)(22,26)(23,27)(24,25)(28,34)(29,32)(30,35)(31,33);(4,8,10,16)(5,12,14,19)(6,7,17,18)(9,13,15,11)(20,22,24,29)(21,26,25,32)(23,30,31,28)(27,35,33,34);(4,7)(5,13)(6,8)(9,12)(10,18)(11,14)(15,19)(16,17)(20,23,24,31)(21,27,25,33)(22,28,29,30)(26,34,32,35);(4,9)(5,6)(7,12)(8,13)(10,15)(11,16)(14,17)(18,19)(20,21)(22,26)(23,27)(24,25)(28,34)(29,32)(30,35)(31,33)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55936);