Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8247 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C7×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,21,24,22,18,20,26)(19,23,28,30,29,25,27,31);(9,12,14,13)(16,18,22,17)(19,25,29,23)(20,24,21,26)(27,30,28,31);(16,19,21,28,22,29,20,27)(17,23,24,30,18,25,26,31)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1235);