Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×C22⋊D7 |
| unambiguous semidirect product | C23⋊D7 |
| Fitting | C23×C7 ⋊ C2 |
| Sylow | C7 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D7 |
| commutator, cosocle | C2×C7 | C23 |
| Frattini | C2 | C22×D7 |
| Fitting | C23×C7 | C2 |
| socle | C22×C7 | C22 |
| hypercenter | C23 | D7 |
| hypocenter | C7 | C2×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
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:
(4,6)(7,8,10,12,13,11,9);(3,4)(5,6)(8,9)(10,11)(12,13);(1,2)34 conjugacy classes — view the full character table.
| maximal subgroup of | #6380, (C22⋊D7)⋊C4, #6445, … |
| maximal normal subgroup of | #6380, (C22⋊D7)⋊C4, #6445, … |
| maximal quotient of | C2×Dic7⋊C4, #6487, C2×(C2×D7)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 36);