Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C4×C7×#471 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#121 |
| prime radical | C2×C7×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4)(21,22,23,24,25,26,27);(6,10)(7,13)(8,16)(12,18)(14,15)(17,19);(5,6,8,10,9,12,16,18)(7,11,14,17,15,20,13,19);(5,7)(6,11)(8,14)(9,15)(10,17)(12,20)(13,16)(18,19)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14540);