Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| automorphism | Aut(G) = C2×S4×AGL(1,𝔽7) (order 2016) · Inn(G) = C22×D7 · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×Q8 ⋊ C2 |
| Sylow | C7 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D7 |
| commutator, socle, cosocle | C2×C7 | C23 |
| Fitting | C7×Q8 | C2 |
| hypercenter | Q8 | D7 |
| hypocenter | C7 | #41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(8,10,11,14)(9,12,13,15);(8,9,11,13)(10,15,14,12);(2,3)(4,5)(6,7)(10,14)(12,15)Polycyclic presentation:
2^4,7|b^2=d,c^2=d,c^a=cd,e^a=e^6,c^b=cd25 conjugacy classes — view the full character table.
| maximal subgroup of | #6477, #6479, #6481, … |
| maximal normal subgroup of | #6477, #6479, #6481, … |
| maximal quotient of | #6453, #6455, #6456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 34);