Finite Group DatabasebetaFinite Group Databasebeta| order | 728 |
| isoclinism | isoclinic class of C7⋊#261 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C7⋊#261 |
| unambiguous semidirect product | (C2×C7)⋊#261 |
| unambiguous semidirect product | D13⋊Dic7 |
| Fitting, socle | C2×C7×C13 ⋊ C4 |
| derived, Hall | C7×C13 ⋊ C2×C4 |
| Sylow | C13 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ C2×#261 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C7⋊#261 |
| commutator, hypocenter | C7×C13 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C13 | C4 |
| prime radical | C2×C7×D13 | C2 |
| cosocle | C7×D13 | C22 |
| LCS | G ▷ C7×C13 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 182 |
62 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×#261, (C2×D7)⋊#261, Dic7⋊#261, … |
| maximal normal subgroup of | Dic7×#261, (C2×D7)⋊#261, Dic7⋊#261, … |
| maximal quotient of | D13⋊(C7⋊C8), #13089849, C4×C7⋊#261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(728, 35);