Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C23×C3×#1020 (order 2304) · Inn(G) = C22 · Out(G) = C23×C3×S4 |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C4×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22 |
| commutator | C2 | C22×C4×C7 |
| Frattini, cosocle | C22 | C23×C7 |
| socle, prime radical | C22×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4)(13,14,15,16,17,18,19);(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, 154);