Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = D4×S4×AGL(1,𝔽7) (order 8064) · Inn(G) = C23×D7 · Out(G) = C22×C3×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#41 ⋊ C2 |
| Sylow | C7 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D7 |
| commutator, socle, cosocle | C2×C7 | C24 |
| Fitting | C7×#41 | C2 |
| hypercenter | #41 | D7 |
| hypocenter | C7 | H(5,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,4,6,7,5,3)(8,10,11,14)(9,13,12,15);(9,12)(13,15);(8,9)(10,13)(11,12)(14,15);(2,3)(4,5)(6,7)(10,14)(13,15)Polycyclic presentation:
2^5,7|d^2=e,d^a=de,f^a=f^6,c^b=ce47 conjugacy classes — view the full character table.
| maximal subgroup of | #89594, #89597, #89970, … |
| maximal normal subgroup of | #89594, #89597, #89970, … |
| maximal quotient of | #90210, #90212, #90214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 185);