Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = AGL(1,𝔽7)×#87524 (order 16128) · Inn(G) = C22×D7 · Out(G) = C3×GL(2,ℤ/4) |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C7 ⋊ C2 |
| Sylow | C7 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C23 |
| Fitting | C24×C7 | C2 |
| hypercenter | C24 | D7 |
| hypocenter | C7 | #120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C24 | |
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);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)Polycyclic presentation:
2^5,7|b^a=be,c^a=cd,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | #89804, #90181, #90185, … |
| maximal normal subgroup of | #89804, #90181, #90185, … |
| maximal quotient of | #89982, #89985, #89989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 148);