Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,14,22,23)(15,24)(16,17)(18,25)(20,21,26,27);(12,13,15,14,18,21,24,20,19,22,17,23,25,27,16,26);(9,11)(12,14)(13,17)(15,22)(16,27)(18,26)(19,23)(20,25)(21,24)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6455);