Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#23578 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1176 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)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,26)(20,28)(21,24)(23,25)(27,30)(29,31);(4,5)(6,7)(9,12)(13,15)(16,18,21,20)(17,22,24,28)(19,25,27,29)(23,30,31,26);(5,7)(16,19,21,27)(17,23,24,31)(18,25,20,29)(22,30,28,26);(8,10,11,14)(9,13,12,15)(19,27)(23,31)(25,29)(26,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16566);