Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#33706 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| prime radical | C22×C7×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
598 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)(18,30)(19,24)(20,38)(21,41)(22,42)(23,29)(25,32)(26,31)(27,40)(28,39)(33,45)(34,44)(35,43)(36,47)(37,46);(9,12)(13,15)(16,18,31,37,22,34,40,35,23,32,21,33,39,36,20,19)(17,24,38,47,28,45,41,25,29,43,27,44,42,46,26,30);(8,10,11,14)(9,13,12,15)(16,19,20,36,39,33,21,32,23,35,40,34,22,37,31,18)(17,25,26,45,42,47,27,24,29,30,41,46,28,44,38,43);(18,32)(19,35)(24,43)(25,30)(33,37)(34,36)(44,47)(45,46)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26694);