Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63051 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6720 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)(6,14)(7,9)(8,12)(10,11)(13,17)(15,19)(16,18)(20,21)(22,27)(23,26)(24,28)(25,30)(29,34)(31,32)(33,35);(5,11)(7,17)(12,14)(15,18)(20,21)(22,27)(23,31)(24,28)(25,30)(26,32)(29,33)(34,35);(4,6,10,8)(5,12,11,14)(7,15,17,18)(9,16,13,19)(20,22)(21,25)(23,29)(24,30)(26,33)(27,28)(31,34)(32,35);(4,7)(5,13)(6,15)(8,18)(9,11)(10,17)(12,19)(14,16)(20,23)(21,26)(22,29)(24,32)(25,33)(27,34)(28,31)(30,35);(4,8,10,6)(5,12,11,14)(7,18,17,15)(9,16,13,19)(20,22)(21,27)(23,29)(24,30)(25,28)(26,34)(31,33)(32,35);(4,9,10,13)(5,7,11,17)(6,16,8,19)(12,15,14,18)(20,21)(22,25)(23,26)(24,28)(27,30)(29,33)(31,32)(34,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56039);