Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#8998 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C7×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,17,21,23)(11,20,22,14)(12,15)(13,16)(18,19)(24,25)(26,32,34,39)(27,29)(28,31)(30,33,37,36)(35,38);(8,10,18,20,13,21,12,14)(9,11,19,17,16,22,15,23)(25,29)(28,35)(30,37)(33,36);(8,11)(9,14)(10,19)(12,23)(13,22)(15,21)(16,20)(17,18)(24,26)(25,30)(27,34)(28,36)(29,37)(31,39)(32,38)(33,35)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1986);