Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#23900 (order 10752) · Inn(G) = C22×D7 · Out(G) = C3×D42 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C7×Q8 ⋊ C2 |
| Sylow | C7 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C23 |
| Fitting | C2×C7×Q8 | C2 |
| prime radical | C2×D28 | C2 |
| hypercenter | C2×Q8 | D7 |
| hypocenter | C7 | #124 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(12,14,15,19)(13,17,16,18);(12,13,15,16)(14,18,19,17);(2,3)(4,5)(6,7)(8,9)(10,11)(13,16)(17,18)Polycyclic presentation:
2^5,7|b^2=e,c^2=e,b^a=be,c^a=cd,f^a=f^6,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #89272, #89525, #89567, … |
| maximal normal subgroup of | #89272, #89525, #89567, … |
| maximal quotient of | #89749, #89758, #89760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 142);