Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C2×C7×#2955 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C23×C7×C8 ◁ G | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(35,36,37,38,39,40,41);(3,4,8,13,9,12,7,21)(5,14,16,10,17,25,22,24)(6,11,19,27,20,26,18,31)(15,28,29,23,30,34,32,33);(3,5)(4,10)(6,15)(7,22)(8,16)(9,17)(11,23)(12,24)(13,25)(14,21)(18,32)(19,29)(20,30)(26,33)(27,34)(28,31);(3,6,8,19,9,20,7,18)(4,11,13,27,12,26,21,31)(5,15,16,29,17,30,22,32)(10,23,25,34,24,33,14,28);(4,12)(10,24)(11,26)(13,21)(14,25)(23,33)(27,31)(28,34)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53167);