Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#17517 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C3×#3481 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×#41 ◁ G | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,10,8,6,11,9)(12,13)(14,18)(15,17)(16,20)(19,23)(21,25)(22,24)(26,27);(4,6)(5,8)(7,9)(10,11)(13,17)(15,23)(18,24)(19,20)(22,27)(25,26);(12,14)(13,18)(15,22)(16,21)(17,24)(19,26)(20,25)(23,27);(14,21)(18,25)(22,27)(24,26)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10505);