Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C7×#1437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C7 | C23 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C26×C7 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25×C7 ◁ G | |
5230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(4,6)(7,8,10,9)(12,13)(15,16,18,17);(11,12,14,13)(15,16,18,17);(3,4,5,6)(7,9,10,8)(11,13,14,12)(15,17,18,16)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6748);