Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×#60376 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G | |
4824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5)(6,7)(9,14)(10,12)(15,21)(16,20)(17,19)(22,23);(8,9)(10,14)(11,15)(12,20)(13,16)(17,21)(18,22)(19,23);(5,7)(8,10,13,12)(9,14,16,20)(11,17,18,19)(15,21,22,23);(8,11)(9,15)(10,17)(12,19)(13,18)(14,21)(16,22)(20,23);(11,18)(15,22)(17,19)(21,23)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53364);