Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#11011 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C23×C7 | C2×C22⋊C4 |
| prime radical | C2×C7×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(16,17,19,22,20,18,23,21);(9,11)(12,13)(14,15)(16,18)(17,20)(19,21)(22,23);(13,15)(16,17)(18,20)(19,22)(21,23)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3999);