Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#21014 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| socle | C2×C3 | C2×#386 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
1748 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)(6,10,16,23)(7,17,19,11)(8,12)(9,13)(14,21)(15,22,28,33)(18,30,31,24)(20,25)(26,32)(27,29)(34,35);(4,6)(5,10,13,23)(7,14)(8,15)(9,16)(11,21,17,29)(12,22,25,33)(18,26)(19,27)(20,28)(24,32,30,35)(31,34);(4,7)(5,11)(6,14)(8,18)(9,19)(10,21)(12,24)(13,17)(15,26)(16,27)(20,31)(22,32)(23,29)(25,30)(28,34)(33,35);(4,8,9,20)(5,12,13,25)(6,15,16,28)(7,18,19,31)(10,22,23,33)(11,24,17,30)(14,26,27,34)(21,32,29,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14002);