Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#8938 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C23×C7 | C2×C22⋊C4 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,17,18,19)(20,21,23,26,24,22,25,27);(12,13,15,19,16,18,14,17)(20,22,23,27,24,21,25,26);(9,11)(12,14,16,15)(13,17,18,19)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1926);