Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,10)(6,14)(8,21)(12,23)(13,24)(15,31)(16,17)(18,19)(22,26)(25,34)(28,29)(30,32);(4,5,8,10,9,13,21,24)(6,11,16,22,17,27,14,26)(7,12,18,23,20,29,19,28)(15,25,31,33,32,34,30,35);(4,6)(5,11)(7,15)(8,16)(9,17)(10,22)(12,25)(13,27)(14,21)(18,31)(19,30)(20,32)(23,33)(24,26)(28,35)(29,34);(4,7)(5,12)(6,15)(8,18)(9,20)(10,23)(11,25)(13,29)(14,30)(16,31)(17,32)(19,21)(22,33)(24,28)(26,35)(27,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25938);