Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C2×C7×#1404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | #124 | C23×C7 |
| prime radical | C2×C7×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#124 ◁ G | |
510 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4,5,9)(6,15,16,12)(7,11,14,18)(8,17,10,13);(3,4)(5,12)(6,15)(7,11)(8,14)(9,13)(10,17)(16,18)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6555);