Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#7285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #99 |
| commutator | C22×C4 | C42×C7 |
| Frattini, cosocle | C2×C4×D4 | C22×C7 |
| socle | C22×C7 | #342 |
| prime radical | C22×C7×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
324 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,12)(13,14)(16,17,19,24,21,25,20,23)(18,26,27,22,29,30,28,31);(8,9)(10,14,15,12)(11,13)(16,18,20,28,21,29,19,27)(17,22,24,30,25,31,23,26)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 273);