Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25636 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9)(10,14)(11,12)(13,15)(16,17)(18,25)(19,27)(20,23)(21,30)(22,26)(24,29)(28,31);(9,12)(13,14)(16,18,20,26)(17,22,23,25)(19,21,28,29)(24,31,30,27);(4,5,6,7)(16,19,21,26,20,28,29,18)(17,22,24,27,23,25,30,31);(8,10,11,15)(9,13,12,14)(17,23)(22,25)(24,30)(27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18624);