Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×C22⋊Q8 ⋊ C2 |
| Sylow | C7 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C22⋊Q8 | C2 |
| prime radical | C2×C22⋊D7 | C22 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #554 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 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) = 224 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,10)(9,13)(11,14)(12,15)(16,19,20,18)(17,22,21,23);(10,14)(13,15)(16,18,20,19)(17,22,21,23);(9,12)(10,14)(16,17)(18,22)(19,23)(20,21);(2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(17,21)(22,23)Polycyclic presentation:
2^6,7|c^2=e,d^2=e,b^a=bf,g^a=g^6,d^b=df,d^c=def67 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90327, #11743054, #11743055, … |
| maximal normal subgroup of | C2×#90327, #11743054, #11743055, … |
| maximal quotient of | #11731325, #11731334, #11731348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1089);