Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#33580 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,17)(15,24)(16,18)(19,21)(20,22)(23,25)(26,27);(8,9,10,11);(13,17)(14,21)(15,22)(18,25)(19,20)(23,27)(24,26);(13,18)(17,25)(19,26)(20,24)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26568);