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×#11281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #397 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
594 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)(14,23)(15,27)(16,32)(17,21)(18,22)(19,25)(20,30)(24,37)(26,35)(28,42)(29,38)(31,41)(33,36)(34,43)(39,40);(12,14)(13,19)(15,28,30,43)(16,33)(17,25)(18,26)(20,37,27,39)(21,35)(22,36)(23,32)(24,31,40,29)(34,38,42,41);(12,15,16,29,18,30,17,31)(13,20,21,38,22,27,32,41)(14,24,25,28,26,40,33,43)(19,34,23,39,36,42,35,37)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4269);