Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C3×#63090 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| socle | C2×C3 | C24×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×#41 ◁ G | |
10836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5)(6,15)(7,12)(8,14)(9,22)(10,13)(11,17)(16,29)(18,26)(19,27)(20,28)(21,32)(23,25)(24,31)(30,35)(33,34);(4,6)(5,11)(7,16)(8,17)(9,23)(10,18)(12,24)(13,25)(14,26)(15,22)(19,30)(20,31)(21,33)(27,34)(28,35)(29,32);(4,7)(5,12)(6,16)(8,20)(9,21)(10,19)(11,24)(13,27)(14,28)(15,29)(17,31)(18,30)(22,32)(23,33)(25,34)(26,35);(7,19)(12,27)(16,30)(20,21)(24,34)(28,32)(29,35)(31,33);(5,13)(11,25)(12,27)(14,22)(15,26)(24,34)(28,32)(29,35);(4,8,10,9)(5,14,13,22)(6,17,18,23)(7,20,19,21)(11,26,25,15)(12,28,27,32)(16,31,30,33)(24,35,34,29)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56078);