Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60825 |
| 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 | |
2146 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)(10,14)(11,13)(12,15)(17,21)(19,26)(22,27)(23,28)(25,30)(29,31);(8,10,11,15)(9,12,13,14)(17,22)(21,27)(23,29)(28,31);(10,15)(12,14)(16,17)(18,23)(19,27)(20,22)(21,26)(24,29)(25,31)(28,30);(9,13)(12,14)(16,18)(17,23)(19,25)(20,24)(21,28)(22,29)(26,30)(27,31);(18,24)(23,29)(25,30)(28,31)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53813);