Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#21995 |
| 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 | |
658 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)(9,12)(10,14)(16,17,21,25,22,26,20,35)(18,23,30,38,28,37,29,44)(19,31,33,39,34,45,32,24)(27,40,43,46,41,47,42,36);(8,9)(10,13)(11,12)(14,15)(16,18)(17,23)(19,27)(20,29)(21,30)(22,28)(24,36)(25,38)(26,37)(31,40)(32,42)(33,43)(34,41)(35,44)(39,46)(45,47);(9,12)(13,15)(16,19,20,32,22,34,21,33)(17,24,25,31,26,39,35,45)(18,27,29,42,28,41,30,43)(23,36,38,40,37,46,44,47);(8,10,11,14)(9,13,12,15)(18,28)(23,37)(27,41)(29,30)(36,46)(38,44)(40,47)(42,43)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14983);