Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C26×C4 |
| direct product | C7×#20347 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C4 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C22×D4 |
| prime radical | C25×C7 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C24×C4×C7 ◁ G | |
2270 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)(10,11)(12,13,14,15)(16,17)(18,26)(19,23)(20,24)(21,25)(22,28)(27,31)(29,30);(9,11)(16,18,20,29,21,27,19,28)(17,22,23,31,25,30,24,26);(18,27)(22,30)(26,31)(28,29);(16,19,21,20)(17,23,25,24)(18,28,27,29)(22,31,30,26)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13335);