Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×C22⋊C8 ⋊ C2 |
| Sylow | C7 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C7×C22⋊C8 | C2 |
| prime radical | C23×D7 | C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C22⋊C8 | D7 |
| hypocenter | C7 | #407 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C22⋊C8 | |
190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(12,13,14,17,15,16,18,19);(8,9)(10,11);(2,3)(4,5)(6,7)(13,16)(17,19)Polycyclic presentation:
2^6,7|c^2=d,d^2=f,c^a=cf,g^a=g^6,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #11725874, #11725875, #11725876, … |
| maximal normal subgroup of | #11725874, #11725875, #11725876, … |
| maximal quotient of | #11725856, #11725857, #11725874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 260);