Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#97 ⋊ C2 |
| Sylow | C7 ⋊ #435 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C7×#97 | C2 |
| prime radical | C2×D28 | C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | #97 | D7 |
| hypocenter | C7 | #435 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ #97 | |
124 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(12,13,14,17,15,16,18,19);(8,9,10,11)(13,16)(17,19);(2,3)(4,5)(6,7)(9,11)Polycyclic presentation:
2^6,7|b^2=e,c^2=d,d^2=f,b^a=be,g^a=g^6,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #11735926, #11735927, C2×#89478, … |
| maximal normal subgroup of | #11735926, #11735927, C2×#89478, … |
| maximal quotient of | #11725832, #11725833, #11725834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 240);