Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,22,27)(18,23,29,37)(19,32,34,41)(20,26,21,25)(24,35,40,33)(28,42,43,47)(30,39,31,38)(36,45,46,44);(9,12)(13,15)(16,18,22,29)(17,23,27,37)(19,28,34,43)(20,30,21,31)(24,36,40,46)(25,38,26,39)(32,42,41,47)(33,44,35,45);(16,19)(17,24)(18,28)(20,33)(21,35)(22,34)(23,36)(25,32)(26,41)(27,40)(29,43)(30,44)(31,45)(37,46)(38,42)(39,47);(8,10,11,14)(9,13,12,15)(18,29)(23,37)(28,43)(30,31)(36,46)(38,39)(42,47)(44,45);(16,20,22,21)(17,25,27,26)(18,30,29,31)(19,33,34,35)(23,38,37,39)(24,32,40,41)(28,44,43,45)(36,42,46,47)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53557);