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×#38337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1474 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)(22,24,28,33)(23,30,32,27)(29,35,34,31);(5,8)(6,10)(12,13,16,15)(14,17,18,19)(21,26)(24,33)(27,30)(31,35);(4,5,7,8)(6,9,10,11)(12,14)(13,17)(15,19)(16,18)(20,22,25,28)(21,24,26,33)(23,29,32,34)(27,31,30,35);(12,13,16,15)(14,17,18,19)(20,23,25,32)(21,27,26,30)(22,29,28,34)(24,31,33,35);(4,6)(5,9)(7,10)(8,11)(12,15,16,13)(14,19,18,17)(20,24)(21,28)(22,26)(23,31)(25,33)(27,34)(29,30)(32,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 31325);