Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#2846 (order 5376) · Inn(G) = C22×D7 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C23 |
| Fitting | C2×C7×D4 | C2 |
| hypercenter | C2×D4 | D7 |
| hypocenter | C7 | #120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
130 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)(9,11)(13,15);(12,13)(14,15);(2,3)(4,5)(6,7)(8,9)(10,11)Polycyclic presentation:
2^5,7|c^a=cd,f^a=f^6,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #89266, #89270, #89513, … |
| maximal normal subgroup of | #89266, #89270, #89513, … |
| maximal quotient of | #89718, #89719, #89727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 132);