Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×C4⋊Q8 ⋊ C2 |
| Sylow | C7 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×C4⋊Q8 | C2 |
| prime radical | C2×D28 | C22 |
| hypercenter | C4⋊Q8 | D7 |
| hypocenter | C7 | #556 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 212 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)(8,11,12,10)(9,15,14,13)(16,18,19,22)(17,21,20,23);(9,14)(10,11)(16,17,19,20)(18,23,22,21);(8,10,12,11)(9,13,14,15);(2,3)(4,5)(6,7)(8,9)(10,13)(11,15)(12,14)(17,20)(18,22)Polycyclic presentation:
2^6,7|b^2=e,c^2=ef,d^2=f,c^a=cf,d^a=def,g^a=g^6,c^b=ce,d^c=df67 conjugacy classes — view the full character table.
| maximal subgroup of | #11726187, #11726189, #11726191, … |
| maximal normal subgroup of | #11726187, #11726189, #11726191, … |
| maximal quotient of | #11731769, #11731770, #11731778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1185);