Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#100 ⋊ C2 |
| Sylow | C7 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6447 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#100 | C2 |
| socle | C2×C7 | #121 |
| hypercenter | #100 | D7 |
| hypocenter | C7 | #472 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #100 | |
108 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(9,14)(10,17)(11,19)(13,20)(15,16)(21,22);(8,10,11,18,12,17,19,23)(9,13,15,21,16,22,14,20);(2,3)(4,5)(6,7)(8,9)(10,13)(11,15)(12,16)(14,19)(17,22)(18,21)(20,23)Polycyclic presentation:
2^6,7|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^6,c^b=cef,e^b=ef,d^c=df49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89521, #11736148, #11736164, … |
| maximal normal subgroup of | C2×#89521, #11736148, #11736164, … |
| maximal quotient of | #11726149, #11726150, #11726153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 283);