Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60399 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7×D4 ◁ G | |
5128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3,4,5,6,7)(13,19)(20,23);(8,9)(10,13)(11,14)(12,15)(16,20)(17,19)(18,21)(22,23);(8,10)(9,13)(11,16)(12,17)(14,20)(15,19)(18,22)(21,23);(8,11)(9,14)(10,16)(12,18)(13,20)(15,21)(17,22)(19,23);(11,18)(14,21)(16,22)(20,23)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53387);