Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#22506 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C22⋊C4 |
| prime radical | C7×#3475 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(13,15)(16,17,20,23,21,24,19,28)(18,25,27,22,26,31,29,30);(8,9)(10,13)(11,12)(14,15)(18,26)(22,30)(25,31)(27,29);(8,10,11,14)(9,13,12,15)(16,18)(17,22)(19,29)(20,27)(21,26)(23,31)(24,30)(25,28);(9,12)(13,15)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15494);