Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#7442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4×C7 |
| Frattini, cosocle | #382 | C22×C7 |
| prime radical | C23×C7 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C7×#382 ◁ G | |
194 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 46 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)(9,12,13,14)(17,21,22,30,23,25,27,31)(18,24,26,28);(8,9)(10,14)(11,13)(12,15)(16,17,19,27,20,23,29,22)(18,25,24,31,26,21,28,30)217 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 430);