Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#8467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1004 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)(10,14,15,12)(11,13)(17,21,23,26)(22,30,31,29);(8,10)(9,12)(11,15)(13,14)(16,17)(18,22)(19,26)(20,23)(21,28)(24,29)(25,31)(27,30);(8,10)(9,12)(11,15)(13,14)(16,18,20,25)(17,22,23,31)(19,27,28,24)(21,29,26,30)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1455);