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 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D7 |
| commutator, socle, cosocle | C2×C7 | C24 |
| Fitting | C7×#41 | C2 |
| hypercenter | #41 | D7 |
| hypocenter | C7 | 2−1+4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,12,13,11)(9,15,16,20)(10,17,18,19)(14,22,23,21);(8,11,13,12)(9,15,16,20)(10,17,18,19)(14,21,23,22);(8,10,13,18)(9,14,16,23)(11,19,12,17)(15,22,20,21);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,20)(13,16)(17,21)(18,23)(19,22)Polycyclic presentation:
2^5,7|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^6,c^b=ce47 conjugacy classes — view the full character table.
| maximal subgroup of | #89598, #89599, #89973, … |
| maximal normal subgroup of | #89598, #89599, #89973, … |
| maximal quotient of | #90207, #90209, #90210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 186);