Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#7493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1390 |
| commutator | C2×D4 | C42×C7 |
| Frattini, cosocle | C2×#127 | C22×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C4×C7 ◁ C2×C7×#127 ◁ G | |
334 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,17,19,23)(11,16,21,14)(12,15)(13,18,22,20);(8,10)(9,13,15,23)(11,20)(12,18)(14,22,16,17)(19,21)196 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 481);