Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7×C8 · Schur multiplier (H2) = C25 |
| direct product | C7×C8×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C8 | D4 |
| commutator | C4 | C23×C7×C8 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C2×C7×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15);(16,17)(18,25)(19,23)(20,28)(21,24)(22,26)(27,30)(29,31);(16,18)(17,22)(19,26)(20,29)(21,27)(23,30)(24,31)(25,28);(16,19,21,20)(17,23,24,28)(18,26,27,29)(22,30,31,25)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10833);