Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60497 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(13,17)(15,22)(18,24)(19,23)(21,26)(25,27);(4,6,7,11)(5,8,9,10);(12,13,16,19)(14,18,20,25)(15,23,22,17)(21,27,26,24);(12,14)(13,18)(15,21)(16,20)(17,24)(19,25)(22,26)(23,27);(14,20)(18,25)(21,26)(24,27)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53485);