Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13038 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #442 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13,17,22,18,23,16,34,19,24,15,31,33,38,32,20)(14,25,28,21,29,37,27,39,30,42,26,40,36,43,35,41);(8,9,10,11)(13,20,24,34)(15,32)(16,17)(18,33)(21,37,40,43)(22,23,31,38)(25,41,42,39)(26,35)(27,28)(29,36);(12,14,19,30)(13,21,24,40)(15,27,17,35)(16,26,32,28)(18,36,33,29)(20,37,34,43)(22,25,31,42)(23,41,38,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6026);