Finite Group DatabasebetaFinite Group Databasebeta| order | 984 |
| isoclinism | isoclinic class of A4⋊D41 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊D41 |
| unambiguous semidirect product | C41⋊S4 |
| Fitting, socle | C22×C41 ⋊ D3 |
| derived | A4×C41 ⋊ C2 |
| Sylow | C41 ⋊ S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×C41 | C2 |
| Fitting, socle | C22×C41 | D3 |
| LCS | G ▷ A4×C41 | |
| UCS | C1 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 164 |
Polycyclic presentation:
2,3,41,2^2|b^a=b^2,c^a=c^40,d^a=e,e^a=d,d^b=e,e^b=de85 conjugacy classes — view the full character table.
| maximal subgroup of | S4×D41, C2×A4⋊D41, #72244678 |
| maximal normal subgroup of | S4×D41, C2×A4⋊D41, #72244678 |
| maximal quotient of | C41⋊#226, C41⋊GL(2,𝔽3), A4⋊Dic41, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(984, 37);