Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#12798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#99 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C22×C7×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
544 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(9,11)(12,13)(14,20,22,27)(15,23,25,18)(16,19)(17,21)(24,26);(8,9,10,11);(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5786);