Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#12051 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C4×C7×C16 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,20,24,21,18,25,27,19,22,28,30,29,26,23,31);(8,9,11,13)(10,15,14,12)(17,22)(18,26)(24,30)(27,31);(16,18,20,27,21,22,25,30,19,26,28,31,29,17,23,24)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5039);