Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C7×C8 |
| Frattini, cosocle | #390 | C22×C7 |
| prime radical | C22×C7×D4 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C7×#390 ◁ G | |
228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17)(18,25,26,21)(19,23,28,22)(20,24)(27,30,31,29);(16,18,19,26,20,27,28,31)(17,21,23,25,24,29,22,30)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 363);