Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#61049 |
| 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 | |
3074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,20)(15,24)(16,18)(17,22)(19,25)(21,27)(23,26);(12,14)(13,17)(15,21)(16,23)(18,27)(19,26)(20,24)(22,25);(9,11);(14,21)(15,25)(17,26)(19,24)(20,27)(22,23);(13,18)(17,27)(19,24)(20,26)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54037);