Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C7 · Schur multiplier (H2) = C214 |
| direct product | C7×#62812 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| socle | C22×C7 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
11346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27);(8,9,10,11);(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(14,21)(17,20)(22,27)(24,26);(13,18)(17,20)(19,25)(24,26);(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 55800);