Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#24693 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5,6,7)(8,9,11,12)(10,14,15,13)(16,17,21,25)(18,22,28,31)(19,23,20,24)(26,29,27,30);(9,12)(13,14)(16,18)(17,22)(19,27)(20,26)(21,28)(23,30)(24,29)(25,31);(5,7)(18,26)(19,20)(22,29)(23,24)(27,28)(30,31);(8,10,11,15)(9,13,12,14)(16,19,21,20)(17,23,25,24)(18,27,28,26)(22,30,31,29)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17681);