Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C23×C7×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C22⋊C4 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C25×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ C24×C4×C7 ◁ G | |
4990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(23,24,25,26,27,28,29);(3,4);(5,6);(7,8,10,14,11,13,19,21)(9,15,16,22,17,12,18,20);(7,9,11,17)(8,12,13,15)(10,18,19,16)(14,22,21,20)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53060);