Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C2×C7×#3019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
5118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4)(5,11)(6,14)(7,9)(8,12)(10,16)(13,18)(15,17);(3,5)(4,8)(6,15)(7,13)(9,17)(10,18)(11,14)(12,16);(6,16)(10,14)(11,18)(12,15);(4,9)(8,17)(10,14)(11,18)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53371);