Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#20829 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C4⋊C4 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,21,25,22,26,20,36)(18,23,30,39,28,38,29,45)(19,31,34,41,35,47,33,24)(27,42,44,40,32,46,43,37);(9,12)(13,15)(16,18)(17,23)(19,32)(20,29)(21,30)(22,28)(24,40)(25,39)(26,38)(27,35)(31,46)(33,44)(34,43)(36,45)(37,41)(42,47);(16,19,20,33,22,35,21,34)(17,24,25,31,26,41,36,47)(18,27,29,43,28,32,30,44)(23,37,39,42,38,40,45,46);(8,10,11,14)(9,13,12,15)(18,28)(23,38)(27,32)(29,30)(37,40)(39,45)(42,46)(43,44)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13817);