Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11258 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C23×C7 | C4×D4 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C23×C7×C8 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,10)(12,13,15,14)(16,17,20,24,21,25,30,31)(18,22,26,28,27,19,23,29);(8,9,11,10)(12,14,15,13)(16,18)(17,22)(19,25)(20,26)(21,27)(23,30)(24,28)(29,31);(8,10,11,9)(13,14)(16,19)(17,23)(18,24)(20,29)(21,22)(25,26)(27,31)(28,30)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4246);