Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
2038 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)(18,28)(19,23)(20,25)(21,36)(22,27)(24,31)(26,29)(30,42)(32,37)(33,38)(34,40)(35,46)(39,44)(41,43)(45,47);(9,12)(13,15)(16,18,29,33,22,31,21,19)(17,23,36,24,27,38,26,28)(20,30,43,45,34,44,35,32)(25,37,46,39,40,47,41,42);(8,10,11,14)(9,13,12,15)(16,19,21,31,22,33,29,18)(17,24,26,23,27,28,36,38)(20,32,35,44,34,45,43,30)(25,39,41,37,40,42,46,47);(16,20)(17,25)(18,30)(19,32)(21,35)(22,34)(23,37)(24,39)(26,41)(27,40)(28,42)(29,43)(31,44)(33,45)(36,46)(38,47);(20,34)(25,40)(30,44)(32,45)(35,43)(37,47)(39,42)(41,46)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53612);