Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#17366 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C22⋊C4 |
| prime radical | C22×C7×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,11,15,12,13,10,14)(16,17)(18,26,29,22)(19,23)(20,24)(21,25)(27,31,28,30);(16,18)(17,22)(19,27)(20,28)(21,29)(23,30)(24,31)(25,26);(9,13)(14,15)(16,19,21,20)(17,23,25,24)(18,27,29,28)(22,30,26,31);(8,10,12,11)(9,14,13,15)(16,20,21,19)(17,24,25,23)(18,28,29,27)(22,31,26,30)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10354);