Finite Group DatabasebetaFinite Group Databasebeta| order | 27300 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C7⋊#13049906) |
| unambiguous semidirect product | C7⋊(C3⋊#13049906) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C7×C13×C25 | C4 |
| Frattini | C5 | C3⋊(C7⋊#63120) |
| socle | C3×C5×C7×C13 | AGL(1,𝔽5) |
| prime radical, cosocle | C3×C7×C13×D25 | C2 |
| LCS | G ▷ C3×C7×C13×C25 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 6825 |
Polycyclic presentation:
2^2,3,5,7,13,5|a^2=b,d^5=g,c^a=c^2,d^a=d^2g,e^a=e^6,f^a=f^12,g^a=g^2,d^b=d^4g^4,g^b=g^41914 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27300, 188);