Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#23668 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G | |
874 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)(12,13)(14,19)(15,17)(16,18)(20,21,25,27,24,29,23,28)(22,26,32,33,30,34,31,35);(8,9)(10,11)(12,14,16,15)(13,17,18,19)(20,22)(21,26)(23,31)(24,30)(25,32)(27,33)(28,35)(29,34);(9,11)(12,15,16,14)(13,17,18,19)(21,27)(23,25)(26,33)(28,29)(31,32)(34,35);(22,30)(26,34)(31,32)(33,35)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16656);