Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#7017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #95 |
| commutator | C2×C4 | C4×C7×C8 |
| Frattini, cosocle | C22×C42 | C22×C7 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G | |
522 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 522 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) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17)(18,22,23,20)(19,21)(24,25,26,28,27,29,30,31);(9,12,14,13)(16,18)(17,20)(19,23)(21,22)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5);