Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61337 |
| 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 ◁ C7×#533 ◁ G | |
1852 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)(13,14)(16,17)(18,21)(19,27)(20,24)(22,26)(23,28)(25,30)(29,31);(8,9,11,12)(10,14,15,13)(16,18,21,17)(19,25,29,23)(20,24,22,26)(27,28,31,30);(16,19)(17,23)(18,25)(20,28)(21,29)(22,30)(24,31)(26,27);(8,10,11,15)(9,13,12,14)(17,18)(23,25)(24,26)(27,31);(16,20,21,22)(17,24,18,26)(19,28,29,30)(23,31,25,27)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54325);