Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#6479 |
| Fitting | C2×C7×SD16 ⋊ C2 |
| Sylow | C7 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×SD16 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×SD16 | D7 |
| hypocenter | C7 | C2×#135 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×SD16 | |
266 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 266 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) = 25 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,6,8,9,7,5)(10,12,14,20)(11,16,15,18)(13,22,21,19)(17,24,23,25);(1,2)(11,15)(12,19)(13,21)(16,24)(18,25)(20,22);(4,5)(6,7)(8,9)(10,11)(12,18)(13,17)(14,15)(16,20)(19,24)(21,23)(22,25);(1,2)70 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#6479, #11737095, #11737119, … |
| maximal normal subgroup of | C4×#6479, #11737095, #11737119, … |
| maximal quotient of | C2×#89528, C2×#89535, C2×#89551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1214);