Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61100 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C2×C3 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,16)(7,18)(8,12)(9,13)(10,14)(11,15)(17,32)(19,21)(20,33)(22,26)(23,35)(24,29)(25,31)(27,30)(28,34);(4,6)(5,7)(8,23)(9,18)(10,26)(11,19)(12,28)(13,21)(14,16)(15,22)(17,24)(20,30)(25,33)(27,34)(29,35)(31,32);(4,7)(5,6)(8,20)(9,21)(10,16)(11,22)(12,17)(13,18)(14,26)(15,19)(23,30)(24,28)(25,35)(27,32)(29,33)(31,34);(4,8)(5,12)(6,17)(7,20)(9,24)(10,27)(11,25)(13,29)(14,30)(15,31)(16,32)(18,33)(19,34)(21,28)(22,35)(23,26);(4,9,11,10)(5,13,15,14)(6,18,19,26)(7,21,22,16)(8,24,25,27)(12,29,31,30)(17,33,34,23)(20,28,35,32)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54088);