Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C7 · Schur multiplier (H2) = C214 |
| direct product | C7×#62654 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C26 |
| commutator | C2 | C25×C4×C7 |
| Frattini, cosocle | C22 | C26×C7 |
| prime radical | C2×C7×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
11222 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)(8,9,10,11)(12,13)(14,18)(15,22)(16,17)(19,23)(20,24)(21,27)(25,26);(13,17)(18,24)(19,22)(25,27);(14,20)(18,24)(21,26)(25,27);(12,14)(13,18)(15,21)(16,20)(17,24)(19,25)(22,27)(23,26);(12,13)(14,18)(15,19)(16,17)(20,24)(21,25)(22,23)(26,27);(12,15,16,23)(13,19,17,22)(14,21,20,26)(18,25,24,27)910 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 55642);