Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#9310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(16,17)(18,22,27,38)(19,29)(20,33)(21,24)(23,31)(25,35)(26,42,30,47)(28,45,44,37)(32,41)(34,40)(36,46,39,43);(8,9,10,15,12,14,11,13)(16,18,21,27)(17,22)(19,30,32,26)(20,28,35,44)(23,39)(24,38)(25,37)(29,47)(31,46,34,43)(33,45)(36,40)(41,42);(8,10,12,11)(9,13,14,15)(16,19,21,32)(17,23,24,40)(18,26,27,30)(20,34,35,31)(22,36,38,39)(25,41,33,29)(28,46,44,43)(37,47,45,42)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2298);