Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×D7 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C16 ⋊ C2 |
| Sylow | C7 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×D7 |
| commutator, socle | C2×C7 | C2×C8 |
| Fitting | C7×C16 | C2 |
| prime radical | C2×D7 | C8 |
| cosocle | C7×C8 | C22 |
| hypercenter | C16 | D7 |
| hypocenter | C7 | M32 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C8 ◁ C16 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,7,5,3)(8,9,10,14,11,15,16,21,12,13,17,19,18,20,22,23);(2,3)(4,5)(6,7)(9,13)(14,19)(15,20)(21,23)Polycyclic presentation:
2^5,7|b^2=c,c^2=d,d^2=e,b^a=be,f^a=f^668 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6372, #89673, D7×M32, … |
| maximal normal subgroup of | C2×#6372, #89673, D7×M32, … |
| maximal quotient of | Dic7⋊C16, #89297, (C2×D7)⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 4);