Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D7×#120 |
| Fitting | C7×#120 ⋊ C2 |
| Sylow | C7 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#120 | C2 |
| hypercenter | #120 | D7 |
| hypocenter | C7 | C2×#120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #120 | |
662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 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 — 4 generators on 15 points:
(1,2,4,6,7,5,3)(13,15);(9,11);(8,9)(10,11)(12,13)(14,15);(2,3)(4,5)(6,7)70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×AΣL(1,𝔽16), D7×B(4,𝔽2), C2×D7×#120, … |
| maximal normal subgroup of | D7×AΣL(1,𝔽16), D7×B(4,𝔽2), C2×D7×#120, … |
| maximal quotient of | #11731263, #11731269, #11731272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1041);