Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×#2234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(12,16)(13,19)(15,22)(17,25)(18,21)(20,26)(23,24);(4,7)(8,10)(11,12,14,16)(13,17,20,24)(15,18,21,22)(19,23,26,25);(3,5,6,9)(4,8,7,10)(11,13,14,20)(12,17,16,24)(15,23,21,25)(18,26,22,19)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26629);