Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#34456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C25 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| prime radical | C2×C7×#546 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
2844 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)(8,9,11,10)(12,13,16,14)(15,18,19,17)(20,21)(22,26)(23,27)(24,25);(13,14)(17,18)(21,25)(22,27);(20,22)(21,23)(24,27)(25,26);(8,10,11,9)(12,14,16,13)(15,18,19,17)(20,22)(21,23)(24,27)(25,26);(12,15)(13,17)(14,18)(16,19)(20,23)(21,22)(24,26)(25,27)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 27444);