Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8267 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C2×D4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C4⋊C4 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
390 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)(17,21,23,26)(22,28,29,30);(8,9,10,13,11,12,14,15)(16,17)(18,22)(19,26)(20,23)(21,27)(24,30)(25,29)(28,31);(9,12)(13,15)(16,18,19,24,20,25,27,31)(17,22,26,30,23,29,21,28)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1255);