Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1306 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)(8,19)(11,20)(12,21)(13,22)(14,26)(15,28)(17,29)(23,33)(24,31)(25,32)(30,34);(4,5,8,10,9,13,19,22)(6,11,15,20,16,24,28,31)(7,12,17,21,18,25,29,32)(14,23,27,30,26,34,35,33);(4,6)(5,11)(7,14)(8,15)(9,16)(10,20)(12,23)(13,24)(17,27)(18,26)(19,28)(21,30)(22,31)(25,34)(29,35)(32,33);(4,7)(5,12)(6,14)(8,17)(9,18)(10,21)(11,23)(13,25)(15,27)(16,26)(19,29)(20,30)(22,32)(24,34)(28,35)(31,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25935);