Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1216 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)(10,13)(11,12)(14,15)(16,17,22,18)(19,27,30,24)(20,23,31,26)(21,28,29,25);(9,12)(13,15)(16,18,22,17)(19,26,30,23)(20,24,31,27)(21,28,29,25);(5,7)(8,10,11,14)(9,13,12,15)(16,19)(17,23)(18,26)(20,29)(21,31)(22,30)(24,25)(27,28);(16,20,22,31)(17,24,18,27)(19,29,30,21)(23,25,26,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26007);