Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60854 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| prime radical | C7×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25)(26,33)(27,31)(28,36)(29,32)(30,35)(34,39)(37,38);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(26,34)(30,38)(33,39)(35,37);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,26)(25,30)(27,35)(28,37)(29,34)(31,39)(32,38)(33,36);(9,15)(13,16)(14,23)(21,22);(24,27,29,28)(25,31,32,36)(26,35,34,37)(30,39,38,33)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53842);