Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#6474 |
| Fitting | C2×C7×D8 ⋊ C2 |
| Sylow | C7 ⋊ C2×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×D8 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×D8 | D7 |
| hypocenter | C7 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(3,4,6,8,9,7,5)(10,11)(12,17)(13,14)(15,16);(1,2)(11,15)(12,16)(14,17);(4,5)(6,7)(8,9)(11,14)(15,17);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#6474, #11737032, #11737048, … |
| maximal normal subgroup of | C4×#6474, #11737032, #11737048, … |
| maximal quotient of | C2×#89529, C2×#89557, C2×#89533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1208);