Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#20708 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#117 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#381 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×#117 ◁ G | |
1260 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)(10,18,20,14)(11,22,24,37)(12,17)(13,16)(15,23,28,38)(19,32)(21,31,35,27)(25,29)(26,34)(30,39)(33,36);(8,10)(9,14)(11,19)(12,21)(13,20)(15,26)(16,18)(17,27)(22,36)(23,34)(24,33)(25,35)(28,30)(29,31)(32,37)(38,39);(8,11)(9,15,16,28)(10,19)(12,23)(13,24)(14,26,18,30)(17,22,29,37)(20,33)(21,34)(25,38)(27,36,31,32)(35,39);(10,20)(14,18)(19,33)(21,35)(26,30)(27,31)(32,36)(34,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13696);