Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#9840 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #127 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5,6,7)(8,9)(10,13)(11,15)(12,14)(16,17)(18,24)(19,27)(20,22)(21,25)(23,28)(26,30)(29,31);(5,7)(8,9,13,15,12,14,11,10)(16,18,19,25,20,26,28,31)(17,21,23,24,22,29,27,30);(8,10,13,9,12,15,11,14)(17,22)(21,29)(23,27)(24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2828);