Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = D7 · Out(G) = C3×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×D7 |
| Fitting | C2×C4×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D7 |
| commutator, hypocenter | C7 | C22×C4 |
| Frattini | C2 | C22×D7 |
| Fitting | C2×C4×C7 | C2 |
| socle | C22×C7 | C22 |
| prime radical | C22×D7 | C2 |
| cosocle | C2×C7 | C23 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(8,9)(10,11)(12,13);(3,4,5,6)(7,8,10,12,13,11,9);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊C8, C42×D7, #6435, … |
| maximal normal subgroup of | (C2×D7)⋊C8, C42×D7, #6435, … |
| maximal quotient of | C4×Dic14, C42×D7, #6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 28);