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 ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C4×C7×D4 | C2 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #553 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 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)(16,19,20,18)(17,21,22,23);(9,13)(10,14)(17,22)(21,23);(8,10,11,14)(9,12,13,15)(16,18,20,19)(17,21,22,23);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^6,7|b^2=f,d^2=ef,c^a=cf,d^a=def,g^a=g^6,c^b=ce85 conjugacy classes — view the full character table.
| maximal subgroup of | #11736859, #11736861, #11736886, … |
| maximal normal subgroup of | #11736859, #11736861, #11736886, … |
| maximal quotient of | #11730902, #11730905, #11730907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1018);