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×#60471 |
| 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 | |
2630 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)(6,15)(8,11)(13,22)(14,23)(16,29)(17,18)(20,24)(21,26)(28,33)(30,31)(32,35);(4,5,11,23,9,14,8,10)(6,12,15,26,18,27,17,21)(7,13,24,33,19,28,20,22)(16,25,29,35,30,34,31,32);(4,6)(5,12)(7,16)(8,17)(9,18)(10,21)(11,15)(13,25)(14,27)(19,30)(20,31)(22,32)(23,26)(24,29)(28,34)(33,35);(4,7)(5,13)(6,16)(8,20)(9,19)(10,22)(11,24)(12,25)(14,28)(15,29)(17,31)(18,30)(21,32)(23,33)(26,35)(27,34);(7,19)(13,28)(16,30)(20,24)(22,33)(25,34)(29,31)(32,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53459);