Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11309 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #387 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1443 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,21,14)(11,15,23,30)(12,17)(13,16)(19,34)(20,35,38,28)(22,39)(24,31,25,33)(26,32)(27,36)(29,37);(8,10)(9,14)(11,22,23,37)(12,20)(13,21)(15,29,30,39)(16,18)(17,28)(19,24,36,25)(26,38)(27,31,34,33)(32,35);(8,11)(9,15,16,30)(10,19)(12,25)(13,23)(14,27,18,34)(17,33,32,31)(20,37)(21,36)(22,38)(24,26)(28,39,35,29)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4297);