Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C7⋊C8 |
| unambiguous semidirect product | (C2×C7)⋊C8 |
| derived, Sylow | C7 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D7 |
| commutator, hypocenter | C7 | C2×C8 |
| Frattini | C4 | C2×D7 |
| Fitting | C2×C4×C7 | C2 |
| socle, prime radical | C22×C7 | C4 |
| cosocle | C4×C7 | C22 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C4 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(4,5)(6,7)(8,9)(10,11,12,14,13,15,16,17);(1,2)(3,4,6,8,9,7,5)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C7⋊C8, #6377, C4⋊(C7⋊C8), … |
| maximal normal subgroup of | C4×C7⋊C8, #6377, C4⋊(C7⋊C8), … |
| maximal quotient of | C4×C7⋊C8, C4⋊(C7⋊C8), C2×C7⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 8);