Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #394 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
606 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)(8,9,10,11)(12,13)(14,18,23,30)(15,19,26,33)(16,21)(17,20)(22,29)(24,31,37,42)(25,32)(27,34,39,40)(28,35)(36,41)(38,43);(12,14)(13,18,20,30)(15,25,26,38)(16,24)(17,23)(19,32)(21,31,35,42)(22,27,36,39)(28,37)(29,40)(33,43)(34,41);(12,15)(13,19,20,33)(14,22,23,36)(16,27)(17,26)(18,29)(21,34,35,40)(24,25,37,38)(28,39)(30,41)(31,32)(42,43)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4273);