Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25799 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,19)(8,23)(9,13)(10,17)(11,21)(12,20)(15,28)(16,29)(18,27)(22,32)(24,31)(25,33)(26,30)(34,35);(4,6,8,17,9,18,20,28)(5,10,12,14,13,15,23,27)(7,16,21,30,22,31,33,35)(11,24,19,26,25,29,32,34);(4,5)(6,15,18,10)(7,11)(8,12)(9,13)(14,17,27,28)(16,29,31,24)(19,21)(20,23)(22,25)(26,30,34,35)(32,33);(4,7,20,33,9,22,8,21)(5,11,23,32,13,25,12,19)(6,16,28,35,18,31,17,30)(10,24,27,34,15,29,14,26)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18787);