Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×#104 ⋊ C2 |
| Sylow | C7 ⋊ #453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6445 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#104 | C2 |
| socle | C2×C7 | #120 |
| hypercenter | #104 | D7 |
| hypocenter | C7 | #453 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #104 | |
168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(9,12,13,14);(8,9)(10,12)(11,13)(14,15);(2,3)(4,5)(6,7)(10,15)(12,14)Polycyclic presentation:
2^6,7|c^2=d,e^2=f,c^a=cd,e^a=ef,g^a=g^6,c^b=ce,d^b=df,e^b=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89594, #11736431, #11736438, … |
| maximal normal subgroup of | C2×#89594, #11736431, #11736438, … |
| maximal quotient of | #11725940, #11725942, #11725948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 356);