Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#7255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #327 |
| commutator | C2×C8 | C42×C7 |
| Frattini, cosocle | C22×M16 | C22×C7 |
| prime radical | C24×C7 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×M16 ◁ G | |
324 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(13,18)(14,22)(15,16)(19,23)(20,27)(21,26)(24,25);(12,13,15,19,17,21,16,20)(14,23,24,26,25,27,22,18)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 243);