Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#11671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C22×C3 | C2×#102 |
| prime radical | C3×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,7,10,8,6,11,9)(12,13)(14,17)(15,24)(16,20)(18,21)(19,22)(23,26)(25,27);(4,6)(5,8)(7,9)(10,11)(13,17)(14,21)(15,22)(18,19)(20,26)(23,27)(24,25);(5,6)(9,10)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4659);