Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1518 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)(4,5)(6,11)(7,9)(8,10)(14,18)(16,19)(20,21,25,22)(23,28,31,26)(24,27,32,29)(30,34,35,33);(4,5,9,7)(6,10,8,11)(12,13,15,17)(14,16,18,19)(21,22)(26,28)(27,29)(33,34);(4,6,9,8)(5,10,7,11)(12,14)(13,16)(15,18)(17,19)(20,22,25,21)(23,28,31,26)(24,29,32,27)(30,34,35,33);(4,7,9,5)(6,10,8,11)(20,23,25,31)(21,26,22,28)(24,30,32,35)(27,33,29,34);(4,8,9,6)(5,10,7,11)(20,24,25,32)(21,27,22,29)(23,30,31,35)(26,33,28,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34390);