Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33736 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| 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 | |
558 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)(6,9)(7,8)(10,11)(13,17)(14,21)(15,22)(18,23)(19,27)(20,26)(24,25);(5,8)(9,11);(12,13,18,27,15,19,14,17,16,20,21,25,22,24,23,26);(4,6,7,10)(5,9,8,11)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26724);