Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1132 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/8)×AGL(1,𝔽7) (order 2688) · Inn(G) = D28 · Out(G) = C24×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)(8,10,12,19,13,20,11,18)(9,14,15,17,16,22,21,23);(2,3)(4,5)(6,7)(8,9,11,15,13,16,12,21)(10,17,18,22,20,23,19,14)Polycyclic presentation:
2^5,7|a^2=c,b^2=d,c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^662 conjugacy classes — view the full character table.
| maximal subgroup of | #89282, #89288, #89301, … |
| maximal normal subgroup of | #89282, #89288, #89301, … |
| maximal quotient of | #89252, #89253, #89347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 25);