Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#32701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2760 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
628 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)(6,14,17,10)(7,18,21,11)(8,12)(9,13)(15,28)(16,24,30,26)(19,23)(20,31,33,25)(22,27)(29,34)(32,35);(4,6)(5,10)(7,19)(8,16)(9,17)(11,23)(12,26)(13,14)(15,21)(18,28)(20,32)(22,30)(24,27)(25,35)(29,33)(31,34);(4,7)(5,11)(6,15,17,19)(8,20)(9,21)(10,23,14,28)(12,25)(13,18)(16,29,30,32)(22,33)(24,34,26,35)(27,31);(4,8,9,22)(5,12,13,27)(6,16,17,30)(7,20,21,33)(10,24,14,26)(11,25,18,31)(15,29,19,32)(23,34,28,35)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25689);