Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, prime radical, cosocle | C22×C7 | C24 |
| Fitting | C7×#125 | C2 |
| hypercenter | #125 | D7 |
| hypocenter | C7 | #557 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #125 | |
200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 448 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,12,15)(9,13,10,14)(16,19)(17,21)(18,22)(20,23);(8,10,12,9)(11,13,15,14)(16,18)(17,19)(20,21)(22,23);(16,18,20,21)(17,19,22,23);(2,3)(4,5)(6,7)(8,9,12,10)(11,14,15,13)(16,17,20,22)(18,19,21,23)Polycyclic presentation:
2^6,7|a^2=f,b^2=e,c^2=ef,d^2=ef,d^a=def,g^a=g^6,c^b=ce,d^c=df67 conjugacy classes — view the full character table.
| maximal subgroup of | #11738596, #11738598, #11738600, … |
| maximal normal subgroup of | #11738596, #11738598, #11738600, … |
| maximal quotient of | #11731602, #11731603, #11731614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1138);