Finite Group DatabasebetaFinite Group Databasebeta| order | 4504 |
| automorphism | Aut(G) order 2531248 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D2252 |
| unambiguous semidirect product | C4⋊D563 |
| Fitting | C4×C563 ⋊ C2 |
| Sylow | C563 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D563 |
| commutator, socle, cosocle | C2×C563 | C22 |
| Fitting | C4×C563 | C2 |
| hypercenter | C4 | D563 |
| hypocenter | C563 | D4 |
| LCS | G ▷ C2×C563 ▷ C563 | |
| UCS | C1 ◁ C2 ◁ C4 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2252 |
Polycyclic presentation:
2^3,563|b^a=bc,d^a=d^562,d^b=d^5621129 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D6756, D11260, D15764, … |
| maximal quotient of | D6756, D11260, D15764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4504, 10);