Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×B(4,𝔽2) (order 384) · Inn(G) = C22 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×D4 |
| unambiguous semidirect product | C4⋊(C22×C7) |
| unambiguous semidirect product | C22⋊(C22×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
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)(7,8,9,10,11,12,13);(3,4,5,6);(4,6)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 38);