Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C2×C7×#2880 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C23×C7×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
5206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(23,24,25,26,27,28,29);(3,4)(5,6)(7,8,10,13,11,14,17,20)(9,12,16,19,15,18,21,22);(4,6);(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22);(9,15)(12,18)(16,21)(19,22)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53051);