Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
2918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(4,6)(7,8)(9,17)(10,16)(11,13)(12,15)(14,18)(19,22)(20,21);(3,4,5,6)(8,13)(14,21)(15,16)(17,22);(7,9)(8,14)(10,18)(11,19)(12,20)(13,21)(15,17)(16,22);(7,10,11,12)(8,15,13,16)(9,18,19,20)(14,17,21,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53402);