Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×D4×D7 |
| Fitting | C22×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D7 |
| commutator, cosocle | C2×C7 | C25 |
| Frattini | C2 | C24×D7 |
| Fitting | C22×C7×D4 | C2 |
| socle | C23×C7 | C23 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | C23×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(3,4)(10,11)(12,13)(14,15);(5,6)(7,8)(9,10,12,14,15,13,11);(6,8);(3,4);(1,2)100 conjugacy classes — view the full character table.
| maximal subgroup of | #11727789, #11727833, #11727952, … |
| maximal normal subgroup of | #11727789, #11727833, #11727952, … |
| maximal quotient of | C22×(C22⋊D7)⋊C4, C22×#6441, C22×#6448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1369);