Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of #13183544 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C7 |
No aliases for this group.
| derived, Hall | #89205 ⋊ C4 |
| Sylow | C72 ⋊ Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | #89205 | C4 |
| Frattini | C3 | #10596885 |
| Fitting, socle | C3×C72 | Dic3 |
| prime radical | C3×C7⋊D7 | D3 |
| cosocle | #11725078 | C2 |
| LCS | G ▷ #89205 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2^2,3^2,7^2|a^2=b,c^3=d,c^a=c^2d^2,d^a=d^2,e^a=e^5f^2,f^a=ef^2,e^b=e^6,f^b=f^6,e^c=e^4f^4,f^c=f^224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 64);