Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4×C7×D4 ⋊ C2 |
| Sylow | C7 ⋊ #529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Frattini | C22 | C23×D7 |
| Fitting | C4×C7×D4 | C2 |
| prime radical | #6502 | C2 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #529 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
234 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(9,13)(10,14)(17,21)(20,23);(16,18,19,22)(17,20,21,23);(8,10)(9,12)(11,14)(13,15);(2,3)(4,5)(6,7)(8,9,11,13)(10,12,14,15)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^6,7|a^2=f,c^2=e,d^a=df,g^a=g^6,d^b=def88 conjugacy classes — view the full character table.
| maximal subgroup of | #11736845, #11736847, #11736852, … |
| maximal normal subgroup of | #11736845, #11736847, #11736852, … |
| maximal quotient of | #11730882, #11730883, #11730884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 991);