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×#61468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1908 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)(9,12)(13,15)(16,17,20,22)(18,24,25,31)(19,27,29,23)(21,28,30,26);(8,9)(10,13)(11,12)(14,15)(18,25)(21,30)(24,31)(26,28);(16,18,20,25)(17,21,22,30)(19,28,29,26)(23,31,27,24);(9,12)(13,15)(17,22)(21,30)(23,27)(24,31);(8,10,11,14)(9,13,12,15)(17,22)(21,30)(23,27)(24,31)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54456);