Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#29270 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2580 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
612 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)(6,15)(7,19)(8,22)(9,14)(10,17)(11,18)(12,20)(13,23)(16,29)(21,33)(24,31)(25,32)(26,35)(27,34)(28,30);(4,6)(5,10,14,25)(7,16)(8,17)(9,18)(11,13,15,22)(12,24,27,29)(19,28,33,26)(20,30)(21,31)(23,32)(34,35);(4,6)(5,11,14,15)(7,16)(8,17)(9,18)(10,22,25,13)(12,26,27,28)(19,24,33,29)(20,30)(21,31)(23,32)(34,35);(4,7,8,20,9,21,23,34)(5,12,13,19,14,27,22,33)(6,16,17,30,18,31,32,35)(10,24,15,28,25,29,11,26)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22258);