Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1141 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C22⋊D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ #108 |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,11,19,13,20,12,21)(9,14,15,18,17,22,16,23);(2,3)(4,5)(6,7)(8,9,12,16,13,17,11,15)(10,18,19,22,20,23,21,14)Polycyclic presentation:
2^5,7|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bd,f^a=f^6,d^b=de44 conjugacy classes — view the full character table.
| maximal subgroup of | #89520, #89521, #89526, … |
| maximal normal subgroup of | #89520, #89521, #89526, … |
| maximal quotient of | #89274, #89275, #89353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 28);