Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#6553 |
| Fitting | C2×C7×#41 ⋊ C2 |
| Sylow | C7 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D7 |
| commutator, cosocle | C2×C7 | C25 |
| Frattini | C2 | C24×D7 |
| Fitting | C2×C7×#41 | C2 |
| socle | C22×C7 | C24 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 5 generators on 17 points:
(1,2)(3,4,6,8,9,7,5)(10,12,13,16)(11,15,14,17);(10,11)(12,15)(13,14)(16,17);(11,14)(15,17);(4,5)(6,7)(8,9)(12,16)(15,17);(1,2)94 conjugacy classes — view the full character table.
| maximal subgroup of | #11727802, #11727893, #11728059, … |
| maximal normal subgroup of | #11727802, #11727893, #11728059, … |
| maximal quotient of | C2×#90210, C2×#90212, C2×#90214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1376);