Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C16 · Schur multiplier (H2) = C1 |
| direct product | C7×#7456 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #106 |
| commutator | C8 | C2×C7×C16 |
| Frattini, cosocle | C4×C16 | C22×C7 |
| socle | C2×C7 | #1367 |
| prime radical | C22×C7 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×C16 ◁ C4×C7×C16 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,19,14,18,25,35,15,20,23,34,11,27,26,38,16,21,31,37,28,36,24,39,13,32,30,17,29,33,10,22);(8,10,11,23,13,24,14,12,16,26,29,30,15,25,28,31)(9,17,18,22,20,19,27,35,21,34,36,38,32,37,33,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 444);