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) = C22×C42 |
| direct product | C7×#22613 |
| 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 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,16)(15,17)(18,19)(20,21,25,28,26,29,23,24)(22,30,31,34,32,35,33,27);(12,14,15,18)(13,16,17,19)(20,22)(21,27)(23,31)(24,30)(25,33)(26,32)(28,35)(29,34);(13,17)(16,19)(20,23,26,25)(21,24,29,28)(22,31,32,33)(27,30,34,35);(20,24,23,29,26,28,25,21)(22,27,33,35,32,34,31,30)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15601);