Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61254 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3475 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7×D4 ◁ G | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,12,11)(10,14,15,13)(16,17,19,23)(18,25,26,21)(20,22,28,24)(27,30,31,29);(8,10)(9,13)(11,14)(12,15)(16,18)(17,21)(19,26)(20,27)(22,29)(23,25)(24,30)(28,31);(8,11,12,9)(10,14,15,13)(17,22)(20,28)(21,29)(23,24)(25,30)(27,31);(10,15)(13,14)(18,26)(21,25)(27,31)(29,30);(17,23)(21,25)(22,24)(29,30)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54242);