Finite Group DatabasebetaFinite Group Databasebeta| order | 624 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #6129⋊C3 |
| Fitting | D4×C13 ⋊ C2×C3 |
| Hall | #6129 ⋊ C3 |
| Hall | C13⋊C3 ⋊ #41 |
| Sylow | C13 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D13⋊C3 |
| commutator, socle, cosocle | C2×C13 | C23×C3 |
| Fitting | D4×C13 | C2×C3 |
| hypercenter | D4 | D13⋊C3 |
| hypocenter | C13 | C3×#41 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ D4 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^3,3,2,13|b^2=e,c^a=ce,f^a=f^12,c^b=ce,f^d=f^340 conjugacy classes — view the full character table.
| maximal subgroup of | #88116⋊C3, #88165⋊C3, #88166⋊C3, … |
| maximal normal subgroup of | #88116⋊C3, #88165⋊C3, #88166⋊C3, … |
| maximal quotient of | #88131⋊C3, #88132⋊C3, #88139⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(624, 142);