Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18269 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,9,13,16)(7,10,14,17)(12,18,19,15)(21,26)(23,24)(27,34)(28,32)(29,30)(33,35);(4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,19)(17,18)(20,21)(22,27)(23,26)(24,32)(25,28)(29,34)(30,33)(31,35);(4,7)(5,10)(6,12)(8,14)(9,15)(11,17)(13,19)(16,18)(20,22,25,31)(21,27,28,35)(23,30,24,29)(26,33,32,34);(20,23,25,24)(21,26,28,32)(22,29,31,30)(27,34,35,33)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11257);