Finite Group DatabasebetaFinite Group Databasebeta| order | 196 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = D7 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic7 |
| unambiguous semidirect product | C7⋊(C4×C7) |
| derived | C7 ⋊ C4×C7 |
| Sylow | C72 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D7 |
| commutator, hypocenter | C7 | C4×C7 |
| Frattini | C2 | C7×D7 |
| Fitting, socle, prime radical | C2×C72 | C2 |
| cosocle | C2×C7 | C2×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C7 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,18,17,16);(8,9,11,13,14,12,10)70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic7, #87639, #87641, … |
| maximal normal subgroup of | D7×Dic7, #87639, #87641, … |
| maximal quotient of | C7×C7⋊C8, C2×C7×Dic7, C3×C7×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(196, 5);