Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #6400 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#442 (order 2688) · Inn(G) = #1141 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C7×D8 ⋊ C2 |
| Sylow | C7 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #1141 |
| commutator, cosocle | C7×C8 | C22 |
| Frattini | C8 | C2×D7 |
| Fitting | C7×D8 | C2 |
| socle | C2×C7 | D8 |
| hypercenter | D8 | D7 |
| hypocenter | C7 | D16 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
28 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)(9,13)(10,17)(11,18)(14,15)(16,22)(19,23)(20,21);(2,3)(4,5)(6,7)(8,9)(10,13)(11,20)(12,16)(14,17)(15,18)(19,22)(21,23)Polycyclic presentation:
2^5,7|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,c^b=cd,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | D7×D16, #89683, #89686, … |
| maximal normal subgroup of | D7×D16, #89683, #89686, … |
| maximal quotient of | #89284, #89286, #89314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 32);