Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#27634 |
| 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 ◁ C22×C3×C42 ◁ G | |
726 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)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,35)(33,34);(5,8)(9,10)(12,14)(13,16)(15,18)(17,19)(21,25)(26,34)(27,32)(28,35);(14,18)(16,19)(20,22,23,29)(21,26,25,34)(24,33,31,30)(27,35,32,28);(4,6,7,11)(5,9,8,10)(21,25)(26,34)(27,32)(28,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20622);