Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×C8×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | D8 |
| commutator | C8 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C2×C7×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C8 ◁ C7×C82 ◁ G | |
258 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15);(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5028);