Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C42 |
| direct product | C7×#21817 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C22⋊C4 |
| prime radical | C7×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
1282 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)(10,14)(11,13)(12,15)(16,17,20,24,21,19,28,23)(18,22,27,30,26,25,31,29);(8,10,11,15)(9,12,13,14)(16,18)(17,22)(19,25)(20,27)(21,26)(23,29)(24,30)(28,31);(10,15)(12,14)(16,19)(17,21)(18,25)(20,23)(22,26)(24,28)(27,29)(30,31);(9,13)(12,14)(18,26)(22,25)(27,31)(29,30)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14805);