Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D35⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D35⋊S4 |
| unambiguous semidirect product | (C5⋊S4)⋊D7 |
| unambiguous semidirect product | (C7⋊S4)⋊D5 |
| Fitting, socle | C22×C5×C7 ⋊ C2×D3 |
| derived | C5×C7×A4 ⋊ C22 |
| Hall | C5×C7 ⋊ C2×S4 |
| Sylow | C7 ⋊ D5×S4 |
| Sylow | C5 ⋊ D7×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C5×C7×A4 | C22 |
| Fitting, socle | C22×C5×C7 | C2×D3 |
| LCS | G ▷ C5×C7×A4 | |
| UCS | C1 | |
132 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^2,3,5,7,2^2|d^a=d^4,e^a=e^6,c^b=c^2,d^b=d^4,f^b=fg,f^c=g,g^c=fg59 conjugacy classes — view the full character table.
| maximal subgroup of | #72218805, D5×D7×S4 |
| maximal normal subgroup of | #72218805, D5×D7×S4 |
| maximal quotient of | #72218791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 956);