Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ #525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Frattini | C22 | C23×D7 |
| Fitting | C4×C7×D4 | C2 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #525 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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)(9,11)(13,17)(16,19);(12,14,15,18)(13,16,17,19);(8,9)(10,11);(2,3)(4,5)(6,7)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^6,7|c^2=e,d^a=de,g^a=g^6,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #11736866, #11736868, #11736873, … |
| maximal normal subgroup of | #11736866, #11736868, #11736873, … |
| maximal quotient of | #11730883, C4×#6441, #11730925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1000);