Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60399 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×#60401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×D4 ◁ G | |
4854 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)(4,5,9,13)(6,10,16,22)(7,11,18,24)(8,12,19,25)(14,26,28,20)(15,21,29,32)(17,23,30,33)(27,34,35,31);(4,6)(5,10)(7,14)(8,15)(9,16)(11,20)(12,21)(13,22)(17,27)(18,28)(19,29)(23,31)(24,26)(25,32)(30,35)(33,34);(4,7)(5,11)(6,14)(8,17)(9,18)(10,20)(12,23)(13,24)(15,27)(16,28)(19,30)(21,31)(22,26)(25,33)(29,35)(32,34);(4,8)(5,12)(6,15)(7,17)(9,19)(10,21)(11,23)(13,25)(14,27)(16,29)(18,30)(20,31)(22,32)(24,33)(26,34)(28,35);(8,19)(12,25)(15,29)(17,30)(21,32)(23,33)(27,35)(31,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53389);