Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23730 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C22×D4 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
1126 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)(12,13,15,17)(14,18,19,16)(20,21,24,29,25,30,23,28)(22,31,32,27,34,35,33,26);(8,9)(10,11)(12,14,15,19)(13,16,17,18)(20,22)(21,26)(23,32)(24,33)(25,34)(27,30)(28,31)(29,35);(9,11)(20,22)(21,27)(23,32)(24,33)(25,34)(26,30)(28,35)(29,31);(20,23,25,24)(21,28,30,29)(22,32,34,33)(26,31,27,35)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16718);