Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#27690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13,15,17)(14,16,18,19)(20,21)(22,32)(23,27)(24,31)(25,30)(26,29)(28,34)(33,35);(4,6,8,7)(5,9,11,10)(12,14)(13,16)(15,18)(17,19)(20,22,25,23)(21,27,30,32)(24,34,26,33)(28,31,35,29);(14,18)(16,19)(20,23,25,22)(21,28,30,35)(24,34,26,33)(27,31,32,29);(4,7,8,6)(5,10,11,9)(20,24,25,26)(21,29,30,31)(22,33,23,34)(27,35,32,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20678);