Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#12789 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×C4⋊C8 |
| prime radical | C24×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C23×C7×C8 ◁ G | |
454 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)(12,13,16,21,17,20,30,35,18,22,31,36,15,28,29,41)(14,23,26,38,27,33,40,42,24,37,39,43,25,19,32,34);(12,14,15,25,18,24,17,27)(13,19,20,23,22,33,28,37)(16,26,29,32,31,39,30,40)(21,34,35,38,36,42,41,43);(9,11)(14,24)(19,33)(23,37)(25,27)(26,39)(32,40)(34,42)(38,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5777);