Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#47806 |
| 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 | |
1534 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)(16,19)(20,21,25,23)(22,28,29,26)(24,31,32,27)(30,33,35,34);(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(21,23)(22,29)(24,32)(33,34);(5,8)(9,10)(12,14)(13,16)(15,18)(17,19)(20,22,25,29)(21,26,23,28)(24,30,32,35)(27,33,31,34);(4,5)(6,9)(7,8)(10,11)(20,23,25,21)(22,26,29,28)(24,31,32,27)(30,33,35,34);(4,6,7,11)(5,9,8,10)(20,24,25,32)(21,27,23,31)(22,30,29,35)(26,33,28,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 40794);