Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#38343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(14,18)(17,19)(20,21,25,26)(22,28,30,24)(23,31,33,27)(29,32,35,34);(5,8)(6,10)(12,13,16,15)(14,17,18,19)(21,26)(24,28)(27,31)(32,34);(4,5,7,8)(6,9,10,11)(12,14)(13,17)(15,19)(16,18)(20,22)(21,24)(23,29)(25,30)(26,28)(27,32)(31,34)(33,35);(12,13,16,15)(14,17,18,19)(20,23,25,33)(21,27,26,31)(22,29,30,35)(24,32,28,34);(4,6)(5,9)(7,10)(8,11)(12,15,16,13)(14,19,18,17)(20,24)(21,22)(23,32)(25,28)(26,30)(27,29)(31,35)(33,34)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 31331);