Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33392 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#102 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#576 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1154 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)(4,5,8,13,9,15,23,14)(6,11,18,28,16,26,31,29)(7,19,21,12,22,10,20,25)(17,32,34,27,30,24,33,35);(5,10)(7,20)(8,23)(11,24)(12,13)(14,25)(15,19)(17,33)(18,31)(21,22)(26,32)(27,28)(29,35)(30,34);(4,6)(5,11)(7,17)(8,18)(9,16)(10,24)(12,27)(13,28)(14,29)(15,26)(19,32)(20,33)(21,34)(22,30)(23,31)(25,35);(6,16)(11,26)(17,30)(18,31)(24,32)(27,35)(28,29)(33,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26380);