Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#62011 |
| 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×#529 ◁ G | |
1856 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)(10,14)(16,17)(18,27)(19,24)(20,26)(21,25)(22,29)(23,28)(30,31);(8,9)(10,13)(11,12)(14,15)(16,18)(17,22)(19,30)(20,29)(21,28)(23,24)(25,31)(26,27);(16,18)(17,23)(19,28)(20,29)(21,30)(22,24)(25,27)(26,31);(9,12)(13,15)(16,19,20,21)(17,24,26,25)(18,28,29,30)(22,31,27,23);(8,10,11,14)(9,13,12,15)(16,19,20,21)(17,25,26,24)(18,28,29,30)(22,23,27,31)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54999);