Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C42×C7 ◁ G | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,20)(19,21)(22,23);(8,10,11,15)(9,12,13,14);(10,15)(12,14)(17,20)(18,23)(21,22);(9,13)(12,14);(17,21)(20,22)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53786);