Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#6498 |
| Sylow | C7 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D7 |
| commutator, cosocle | C22×C7 | C24 |
| Frattini | C22 | C23×D7 |
| Fitting, prime radical | C22×C7×D4 | C2 |
| socle | C23×C7 | C23 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C2×#123 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 21 points:
(3,4)(5,6)(7,8,10,12,13,11,9)(14,16)(15,18)(17,20)(19,21);(1,2)(16,20)(18,21);(4,6)(8,9)(10,11)(12,13)(14,15,17,19)(16,18,20,21);(1,2)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89267, #11727777, #11727783, … |
| maximal normal subgroup of | C2×#89267, #11727777, #11727783, … |
| maximal quotient of | C2×#89715, C2×#89718, C2×#89716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1249);