Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 55296 |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C22×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(13,14,15,16,17,18,19);(3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 191);