Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60620 |
| 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 | |
2014 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)(18,28,30,39)(19,33,34,41)(20,26,21,25)(23,32,38,31)(24,35,40,36)(29,37,43,47)(42,44,46,45);(9,12)(13,15)(16,18,21,32,22,30,20,31)(17,23,26,39,27,38,25,28)(19,29,36,45,34,43,35,44)(24,37,33,46,40,47,41,42);(8,10,11,14)(9,13,12,15)(16,19,20,35,22,34,21,36)(17,24,25,41,27,40,26,33)(18,29,31,44,30,43,32,45)(23,37,28,42,38,47,39,46);(19,34)(24,40)(29,43)(33,41)(35,36)(37,47)(42,46)(44,45);(18,30)(23,38)(28,39)(29,43)(31,32)(37,47)(42,46)(44,45)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53608);