Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 18432 |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C22×C7×D4 |
| unambiguous semidirect product | C4⋊(C23×C7) |
| unambiguous semidirect product | C22⋊(C23×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(9,10,11,12,13,14,15);(3,4);(5,6,7,8);(6,8)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 190);