Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63026 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6678 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)(4,5,9,11)(6,14,18,28)(7,13,19,25)(8,12,21,24)(10,15,23,17)(16,27,30,35)(20,26,32,34)(22,29,33,31);(4,6,8,17)(5,10,12,14)(7,16,20,31)(9,18,21,15)(11,23,24,28)(13,22,26,27)(19,30,32,29)(25,33,34,35);(5,11)(6,15)(10,14)(12,24)(13,25)(16,29)(17,18)(22,27)(23,28)(26,34)(30,31)(33,35);(5,12)(10,14)(11,24)(13,26)(22,27)(23,28)(25,34)(33,35);(4,7,8,20)(5,13,12,26)(6,16,17,31)(9,19,21,32)(10,22,14,27)(11,25,24,34)(15,29,18,30)(23,33,28,35);(7,19)(13,25)(16,30)(20,32)(22,33)(26,34)(27,35)(29,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56014);