Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of #1141 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C22⋊D7 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D7 |
| commutator, cosocle | C4×C7 | C22 |
| Frattini | C4 | C2×D7 |
| Fitting | C7×Q8 | C2 |
| socle, prime radical | C2×C7 | D4 |
| hypercenter | Q8 | D7 |
| hypocenter | C7 | Q16 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16);(2,3)(4,5)(6,7)(8,9,12,15)(10,16,18,23)(11,19,21,14)(13,20,22,17)Polycyclic presentation:
2^4,7|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^6,c^b=cd22 conjugacy classes — view the full character table.
| maximal subgroup of | #6478, #6479, D7×Q16, … |
| maximal normal subgroup of | #6478, #6479, D7×Q16, … |
| maximal quotient of | #6381, #6384, #6409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 17);