Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2380 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,21,14)(11,22,25,36)(12,16)(13,17)(15,24,32,38)(19,30)(20,33,26,29)(23,28)(27,31)(34,37)(35,39);(8,10)(9,14)(11,23)(12,26)(13,21)(15,30)(16,33)(17,18)(19,24)(20,27)(22,34)(25,37)(28,36)(29,31)(32,39)(35,38);(8,11,12,24,13,25,27,38)(9,15,31,36,17,32,16,22)(10,19,20,23,21,35,26,37)(14,28,33,30,18,34,29,39);(9,16,17,31)(14,29,18,33)(15,22,32,36)(28,39,34,30)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16348);