Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = D42×AGL(1,𝔽7) (order 2688) · Inn(G) = C2×C22⋊D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C22⋊D7 |
| commutator, cosocle | C4×C7 | C23 |
| Frattini | C4 | C22×D7 |
| Fitting | C2×C7×D4 | C2 |
| socle | C2×C7 | C2×D4 |
| hypercenter | C2×D4 | D7 |
| hypocenter | C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D4 | |
68 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)(9,14)(10,11)(13,15);(8,10,12,11)(9,13,15,14);(2,3)(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)Polycyclic presentation:
2^5,7|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^6,c^b=ce,d^c=de41 conjugacy classes — view the full character table.
| maximal subgroup of | #89520, #89521, #89834, … |
| maximal normal subgroup of | #89520, #89521, #89834, … |
| maximal quotient of | #89736, #89738, #89740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 127);