Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C3×#3484 (order 768) · Inn(G) = D4 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C2×C7×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C7 |
| prime radical | C2×C7×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(11,12,13,14,15,16,17);(3,4,5,6,7,8,9,10);(4,6)(5,9)(8,10)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 168);