Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,21)(11,15)(12,20)(13,18)(14,19)(16,23)(17,25)(22,36)(24,35)(26,31)(27,30)(28,34)(29,37)(32,39)(33,38);(9,15)(11,25)(12,13)(16,32)(17,20)(18,31)(19,30)(21,36)(22,37)(23,28)(26,27)(29,34)(33,35)(38,39);(8,10)(9,16)(11,22)(12,28)(13,23)(14,24)(15,29)(17,33)(18,35)(19,34)(20,21)(25,37)(26,39)(27,38)(30,32)(31,36)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6298);