Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C82 · Schur multiplier (H2) = C23×C8 |
| direct product | C7×C8×C22⋊C8 |
| unambiguous semidirect product | C22⋊(C7×C82) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7×C8 | C22 |
| commutator | C2 | C2×C7×C82 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C23×C7 | C2×C42 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7×C8 ◁ G | |
726 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)(8,9,10,11,12,13,14,15);(16,17)(18,19)(20,21,22,24,23,25,26,27);(17,19)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1131);