Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C8 |
| direct product | C7×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C2×D4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C4⋊C4 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
342 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)(9,13,15,18)(14,20,21,22)(24,25)(26,29)(27,31)(28,30)(32,36)(33,35)(34,37)(38,39);(8,9)(10,14)(11,18)(12,15)(13,19)(16,22)(17,21)(20,23)(24,26,27,32,28,33,34,38)(25,29,31,36,30,35,37,39);(8,10,11,16,12,17,19,23)(9,14,18,22,15,21,13,20)(25,30)(29,35)(31,37)(36,39)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1258);