Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#11584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C8 |
| prime radical | C7×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,19,24,20,25,29,37,21,23,30,35,31,36,41,45)(18,22,27,32,28,33,40,42,26,34,38,43,39,44,46,47);(9,12)(10,15)(13,14)(16,18)(17,22)(19,27)(20,28)(21,26)(23,34)(24,32)(25,33)(29,40)(30,38)(31,39)(35,43)(36,44)(37,42)(41,46)(45,47);(17,23)(18,26)(24,35)(25,36)(27,38)(28,39)(37,45)(40,46)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4572);