Finite Group DatabasebetaFinite Group Databasebeta| order | 624 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C13⋊#226 |
| Sylow | C13 ⋊ #226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊D13 |
| commutator, prime radical, cosocle, hypocenter | C13×SL(2,𝔽3) | C2 |
| Fitting | Q8×C13 | D3 |
| socle | C2×C13 | S4 |
| LCS | G ▷ C13×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
Polycyclic presentation:
2,3,13,2^3|a^2=f,d^2=f,e^2=f,b^a=b^2,c^a=c^12,d^a=ef,e^a=df,d^b=e,e^b=def,e^d=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #11929062, #11929063, D13×#226, … |
| maximal normal subgroup of | #11929062, #11929063, D13×#226, … |
| maximal quotient of | #11928231, C2×C13⋊#226, C13⋊#3688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(624, 134);