Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×#63003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
11382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 23 points:
(1,2,3)(5,7)(8,9,13,14)(10,12,16,21)(11,15,20,22)(17,19,18,23);(4,5,6,7);(9,14)(12,21)(15,22)(19,23);(8,10,13,16)(9,12,14,21)(11,18,20,17)(15,23,22,19);(8,11,13,20)(9,15,14,22)(10,17,16,18)(12,19,21,23);(8,12)(9,16)(10,14)(11,19)(13,21)(15,18)(17,22)(20,23)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55991);