Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×H(5,𝔽2) ⋊ C2 |
| Sylow | C7 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×H(5,𝔽2) | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | H(5,𝔽2) | D7 |
| hypocenter | C7 | #576 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 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 — 4 generators on 23 points:
(1,2,4,6,7,5,3)(9,14)(11,19)(13,21)(15,16)(18,23)(20,22);(10,17)(13,20)(18,23)(21,22);(8,10)(9,13)(11,18)(12,17)(14,21)(15,22)(16,20)(19,23);(2,3)(4,5)(6,7)(8,9)(10,13)(11,14)(12,16)(15,19)(17,20)(18,21)(22,23)Polycyclic presentation:
2^6,7|e^2=f,d^a=de,e^a=ef,g^a=g^6,c^b=cf,e^d=ef73 conjugacy classes — view the full character table.
| maximal subgroup of | #11737725, #11737727, #11737739, … |
| maximal normal subgroup of | #11737725, #11737727, #11737739, … |
| maximal quotient of | #11739982, #11739983, #11739984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1288);