Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#32984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(16,17,21,24)(18,25,27,30)(19,23,20,28)(22,29,31,26);(9,12)(13,15);(8,10,11,14)(9,13,12,15)(16,18)(17,22)(19,26)(20,29)(21,27)(23,30)(24,31)(25,28);(5,7)(16,19,21,20)(17,23,24,28)(18,26,27,29)(22,30,31,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25972);