Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | D7×#124 |
| Fitting | C7×#124 ⋊ C2 |
| Sylow | C7 ⋊ C2×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#124 | C2 |
| prime radical | C2×D4×D7 | C2 |
| hypercenter | #124 | D7 |
| hypocenter | C7 | C2×#124 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #124 | |
330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 19 points:
(1,2,4,6,7,5,3)(8,10,11,9)(13,17)(16,19);(8,9,11,10)(12,14,15,18)(13,16,17,19);(9,10)(12,13)(14,16)(15,17)(18,19);(2,3)(4,5)(6,7)70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#329, #11726097, #11726101, … |
| maximal normal subgroup of | D7×#329, #11726097, #11726101, … |
| maximal quotient of | #11731594, #11731609, #11731612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1126);