Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C82 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#7139 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C22⋊C4 |
| commutator | C22 | C7×C82 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| socle | C22×C7 | #336 |
| prime radical | C23×C7 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
262 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 2 generators on 31 points:
(1,2,3,4,5,6,7)(16,17,18,22,19,23,25,29,20,21,24,27,26,28,30,31);(8,9,10,12,11,13,14,15)(17,21)(18,24)(23,28)(25,30)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 127);