Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#10263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9)(10,12)(11,14)(13,15)(16,17,21,25)(18,26,29,30)(19,24,20,23)(22,28,31,27);(5,7)(16,18,21,29)(17,22,25,31)(19,28,20,27)(23,30,24,26);(9,12)(10,15)(13,14)(18,27)(19,20)(22,26)(23,24)(28,29)(30,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3251);