Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C2×C7×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G | |
2206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(17,21)(18,24)(19,26)(22,29)(23,28)(25,30);(9,12)(13,15)(16,17,19,21,20,23,26,28)(18,22,24,27,25,29,30,31);(8,10,11,14)(9,13,12,15)(16,18,19,24,20,25,26,30)(17,22,21,27,23,29,28,31);(18,25)(22,29)(24,30)(27,31);(17,23)(21,28)(22,29)(27,31)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53607);