Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#12982 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #120 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1869 |
| prime radical | C7×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,19,14,20,32,38,15,18,33,37,12,17,22,35)(10,21,25,30,23,36,39,11,24,27,31,26,29,16,34,28);(8,10,22,39,12,29,33,25,15,24,32,34,14,23,13,31)(9,16,35,30,17,27,37,28,18,36,38,26,20,21,19,11);(8,11)(9,10)(12,30)(13,21)(14,26)(15,28)(16,22)(17,29)(18,24)(19,31)(20,23)(25,37)(27,33)(32,36)(34,38)(35,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5970);