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) = C24 |
| direct product | C3×#18037 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C2×C3×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×M16 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,10,8,6,9,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27);(12,14,17,23)(13,18,20,27)(15,22,16,25)(19,26,24,21);(4,6,7,11,8,5,9,10);(5,6)(10,11)(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11025);