Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7159 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C8 | C4×C7×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| socle | C22×C7 | #340 |
| prime radical | C23×C7 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(17,21)(18,25)(19,26)(22,24)(23,30)(27,31)(28,29);(16,17,18,22,26,28,31,29,20,24,27,21,19,23,25,30)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 147);