Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#22571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C4×C7×C22⋊C4 ◁ G | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,16)(15,17)(18,19)(20,21,24,27)(22,23,25,26);(12,14,15,18)(13,16,17,19)(21,25)(22,27)(23,26);(13,17)(16,19);(20,22,24,25)(21,26,27,23)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15559);