Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×#476 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
524 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)(4,5)(6,18)(7,22,25,35)(8,14)(9,16)(10,26)(11,17)(12,20)(13,19,29,34)(15,21)(23,32,24,30)(27,31,33,28);(4,6,9,20,11,21,8,10)(5,12,14,18,17,26,16,15)(7,19,24,33,25,34,23,27)(13,22,28,32,29,35,31,30);(4,7)(5,13)(6,19)(8,23)(9,24)(10,27)(11,25)(12,22)(14,28)(15,30)(16,31)(17,29)(18,32)(20,33)(21,34)(26,35);(4,8,11,9)(5,14,17,16)(6,10,21,20)(7,23,25,24)(12,18,26,15)(13,28,29,31)(19,27,34,33)(22,32,35,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26625);