Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#32896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
3610 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)(4,5,10,16)(6,17,20,30)(7,13,22,29)(8,15,9,14)(11,23,27,19)(12,24,28,21)(18,25,35,32)(26,34,31,33);(4,6)(5,11)(7,18)(8,23)(9,19)(10,20)(12,25)(13,26)(14,30)(15,17)(16,27)(21,34)(22,35)(24,33)(28,32)(29,31);(4,7)(5,12)(6,18)(8,21)(9,24)(10,22)(11,25)(13,14)(15,29)(16,28)(17,31)(19,33)(20,35)(23,34)(26,30)(27,32);(4,7)(5,13)(6,18)(8,24)(9,21)(10,22)(11,26)(12,14)(15,28)(16,29)(17,32)(19,34)(20,35)(23,33)(25,30)(27,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25884);