Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#6500 |
| Fitting | C22×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D7 |
| commutator, cosocle | C22×C7 | C24 |
| Frattini | C22 | C23×D7 |
| Fitting | C22×C7×D4 | C2 |
| socle | C23×C7 | C23 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C2×#120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
662 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 143 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)(4,6)(8,10)(11,12,14,16,17,15,13);(1,2)(3,4)(5,6);(7,8)(9,10)(12,13)(14,15)(16,17);(1,2)88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89266, #11727776, #11727790, … |
| maximal normal subgroup of | C2×#89266, #11727776, #11727790, … |
| maximal quotient of | C2×#89718, C2×#89719, C2×#89727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1252);