Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60376 |
| 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 ◁ C22×C7×#41 ◁ G | |
4824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,18)(14,16)(19,25)(20,24)(21,23)(26,27);(12,13)(14,18)(15,19)(16,24)(17,20)(21,25)(22,26)(23,27);(9,11)(12,14,17,16)(13,18,20,24)(15,21,22,23)(19,25,26,27);(12,15)(13,19)(14,21)(16,23)(17,22)(18,25)(20,26)(24,27);(15,22)(19,26)(21,23)(25,27)427 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53364);