Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×#41 ◁ G | |
2604 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)(5,10)(6,14)(8,20)(12,22)(13,23)(15,28)(16,17)(19,31)(21,25)(27,33)(29,30)(32,35);(4,5,8,10,9,13,20,23)(6,11,16,21,17,26,14,25)(7,12,19,22,18,27,31,33)(15,24,30,32,29,34,28,35);(4,6)(5,11)(7,15)(8,16)(9,17)(10,21)(12,24)(13,26)(14,20)(18,29)(19,30)(22,32)(23,25)(27,34)(28,31)(33,35);(4,7)(5,12)(6,15)(8,19)(9,18)(10,22)(11,24)(13,27)(14,28)(16,30)(17,29)(20,31)(21,32)(23,33)(25,35)(26,34);(7,18)(12,27)(15,29)(19,31)(22,33)(24,34)(28,30)(32,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53458);