Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#7131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #99 |
| commutator | C22×C4 | C2×C7×C8 |
| Frattini, cosocle | C2×#95 | C22×C7 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C24×C7 ◁ C2×C7×#95 ◁ G | |
476 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 90 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)(16,17,18,20,19,21,22,23)(24,25)(26,30,31,28)(27,29);(8,9,11,13)(10,14,15,12)(25,28)(29,30)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 119);