Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C2×C7×#2883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| prime radical | C22×C7×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7×D4 ◁ G | |
5338 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)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17);(3,6)(4,9)(5,12)(7,14)(8,15)(10,16)(11,17)(13,18);(6,14)(9,16)(12,18)(15,17)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53056);