Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11260 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #117 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #382 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
590 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)(12,13,17,23,18,21,16,22)(14,24,26,36,27,19,31,34)(15,28,29,35,30,37,32,20)(25,39,41,38,42,43,40,33);(8,9)(10,11)(12,14,16,26,18,27,17,31)(13,19,22,34,21,24,23,36)(15,25,29,40,30,42,32,41)(20,33,28,43,35,38,37,39);(9,11)(12,15)(13,20,21,35)(14,25)(16,29)(17,32)(18,30)(19,33,24,38)(22,28,23,37)(26,40)(27,42)(31,41)(34,43,36,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4248);