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×#33388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#102 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1128 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)(5,10,12,24)(7,17)(8,20)(11,23,26,34)(13,18,21,25)(15,30)(16,29)(19,22)(27,31,32,35)(28,33);(4,5)(6,11)(7,18)(8,21)(9,12)(10,22)(13,20)(14,26)(15,31)(16,32)(17,24)(19,25)(23,33)(27,29)(28,35)(30,34);(4,6)(5,11)(7,15)(8,16)(9,14)(10,23)(12,26)(13,27)(17,30)(18,31)(19,28)(20,29)(21,32)(22,33)(24,34)(25,35);(6,14)(11,26)(15,28)(16,29)(23,34)(27,32)(30,33)(31,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26376);