Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#124 | C2 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | #124 | D7 |
| hypocenter | C7 | #556 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #124 | |
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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)(10,15)(12,14)(17,21)(20,23);(8,10,11,15)(9,12,13,14)(16,18,19,22)(17,20,21,23);(10,15)(12,14)(16,17)(18,20)(19,21)(22,23);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)Polycyclic presentation:
2^6,7|a^2=f,c^2=e,c^a=cf,g^a=g^6,c^b=cf,d^b=def,d^c=df67 conjugacy classes — view the full character table.
| maximal subgroup of | #11737961, #11737963, #11737967, … |
| maximal normal subgroup of | #11737961, #11737963, #11737967, … |
| maximal quotient of | #11731603, #11731604, #11731618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1124);