Finite Group DatabasebetaFinite Group Databasebeta| order | 12054 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C41⋊#64424 |
| Fitting, socle, derived, Hall | C72×C41 ⋊ C2×C3 |
| Hall | C41×#3861 ⋊ C2 |
| Hall | C7⋊D287 ⋊ C3 |
| Hall | #3861 ⋊ D41 |
| Sylow | C72 ⋊ C3×D41 |
| Sylow | C41 ⋊ #64424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C41 | C2×C3 |
| LCS | G ▷ C72×C41 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7^2,41|c^a=c^6,d^a=d^6,e^a=e^40,c^b=c^2,d^b=d^2394 conjugacy classes — view the full character table.
| maximal subgroup of | #22548818⋊C3, D41×#64424, #64423⋊D41, … |
| maximal normal subgroup of | #22548818⋊C3, D41×#64424, #64423⋊D41, … |
| maximal quotient of | #37563039, C2×C41⋊#64424, C41⋊#11725077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12054, 18);