Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C2×#120×AGL(1,𝔽7) (order 2688) · Inn(G) = C2×D7 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D7 |
| commutator | C2×C7 | C42 |
| Fitting | C2×C7×C8 | C2 |
| socle, prime radical | C22×C7 | C2×C4 |
| cosocle | C2×C4×C7 | C22 |
| hypercenter | C2×C8 | D7 |
| hypocenter | C7 | #97 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,6,7,5,3)(12,13,14,17,15,16,18,19);(2,3)(4,5)(6,7)(8,9,10,11)(13,16)(17,19)Polycyclic presentation:
2^5,7|a^2=c,b^2=d,d^2=e,b^a=be,f^a=f^668 conjugacy classes — view the full character table.
| maximal subgroup of | #89307, #89451, C4×#1131, … |
| maximal normal subgroup of | #89307, #89451, C4×#1131, … |
| maximal quotient of | #89249, #89250, #89307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 21);