Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C43 · Schur multiplier (H2) = C24×C42 |
| direct product | C2×C3×#1726 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C8 | C22 |
| commutator | C2 | C2×C3×C43 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle, prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C8 ◁ G | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(3,4,5,6)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(7,8,9,10)(13,19)(16,22)(20,25)(23,26)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13315);