Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60088 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C22×C7×2−1+4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C7×#41 ◁ G | |
4810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,22)(15,19)(16,28)(17,20)(18,24)(21,30)(23,36)(25,35)(26,33)(27,40)(29,32)(31,38)(34,39)(37,43)(41,42);(12,14,17,25)(13,18,20,32)(15,23,26,37)(16,29,30,24)(19,31,33,42)(21,35,28,22)(27,41,39,38)(34,43,40,36);(12,15)(13,19)(14,23)(16,27)(17,26)(18,31)(20,33)(21,34)(22,36)(24,38)(25,37)(28,40)(29,41)(30,39)(32,42)(35,43);(15,26)(19,33)(23,37)(27,39)(31,42)(34,40)(36,43)(38,41);(13,20)(18,32)(19,33)(21,28)(22,35)(31,42)(34,40)(36,43)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53076);