Finite Group DatabasebetaFinite Group Databasebeta| order | 224 |
| isoclinism | isoclinic class of #1141 |
| automorphism | Aut(G) = C23×D4×AGL(1,𝔽7) (order 2688) · Inn(G) = C22⋊D7 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #103 |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,6,7,5,3)(12,13,15,17)(14,19,18,16);(2,3)(4,5)(6,7)(8,9,10,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^5,7|a^2=c,b^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^6,d^b=de44 conjugacy classes — view the full character table.
| maximal subgroup of | #89563, #89564, #89566, … |
| maximal normal subgroup of | #89563, #89564, #89566, … |
| maximal quotient of | #89324, Q8⋊(C7⋊C8), #89333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(224, 41);