Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C26×C42 |
| direct product | C4×C7×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| socle | C23×C7 | C25 |
| prime radical | C2×C7×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
2664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4)(21,22,23,24,25,26,27);(5,6,9,12)(7,14,15,11)(8,16,18,20)(10,13,19,17);(5,7,9,15)(6,11,12,14)(8,17,18,13)(10,16,19,20);(5,8)(6,7)(9,18)(10,11)(12,15)(13,16)(14,19)(17,20);(5,7)(6,8)(9,15)(10,16)(11,13)(12,18)(14,17)(19,20)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 27341);