Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| automorphism | Aut(G) = D42×AGL(1,𝔽7) (order 2688) · Inn(G) = C2×D28 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×M16 ⋊ C2 |
| Sylow | C7 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D28 |
| commutator, cosocle | C4×C7 | C23 |
| Frattini | C4 | C22×D7 |
| Fitting | C7×M16 | C2 |
| socle | C2×C7 | C2×D4 |
| prime radical | #1157 | C2 |
| hypercenter | M16 | D7 |
| hypocenter | C7 | #137 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,12,19,13,20,10,17)(9,15,14,18,16,23,21,22);(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,22,23,18);(2,3)(4,5)(6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,23)(20,22)Polycyclic presentation:
2^5,7|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^6,c^b=ce41 conjugacy classes — view the full character table.
| maximal subgroup of | #89518, #89520, #89524, … |
| maximal normal subgroup of | #89518, #89520, #89524, … |
| maximal quotient of | #89474, #89475, #89482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 104);