Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3552 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
866 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)(4,5,8,12)(6,10,13,17)(7,9,15,11)(14,16,19,18)(22,28)(23,30)(25,33)(26,31)(29,32)(34,35);(5,9)(7,15)(10,16)(11,12)(14,19)(17,18)(20,21,24,27)(22,25,29,34)(23,31,30,26)(28,35,32,33);(4,6)(5,10)(7,14)(8,13)(9,16)(11,18)(12,17)(15,19)(20,22)(21,25)(23,32)(24,29)(26,35)(27,34)(28,30)(31,33);(6,13)(10,17)(14,19)(16,18)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11683);