Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = AGL(1,𝔽7)×AGL(3,𝔽2) (order 56448) · Inn(G) = D7 · Out(G) = C3×AGL(3,𝔽2) |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C23×D7 |
| unambiguous semidirect product | C7⋊C24 |
| unambiguous semidirect product | (C2×C7)⋊C23 |
| Fitting, socle | C23×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D7 |
| commutator, cosocle, hypocenter | C7 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4)(5,6)(8,9)(10,11)(12,13);(5,6)(7,8,10,12,13,11,9);(3,4);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C22⋊C4, #6445, C2×(C2×D7)⋊C4, … |
| maximal normal subgroup of | D7×C22⋊C4, #6445, C2×(C2×D7)⋊C4, … |
| maximal quotient of | C22×Dic14, C22×C4×D7, C22×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 42);