Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×D4×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11);(9,11);(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27);(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54036);