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×#61177 |
| 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×#525 ◁ G | |
2102 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)(8,9,11,13)(10,14,15,12)(18,26)(19,20)(22,29)(23,25)(27,28)(30,31);(8,10)(9,12)(11,15)(13,14)(16,17)(18,22)(19,23)(20,25)(21,24)(26,29)(27,30)(28,31);(16,18)(17,22)(19,26)(20,28)(21,27)(23,29)(24,30)(25,31);(10,15)(12,14)(16,19,21,20)(17,23,24,25)(18,26,27,28)(22,29,30,31);(17,24)(22,30)(23,25)(29,31)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54165);