Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×2−1+4 ⋊ C2 |
| Sylow | C7 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×2−1+4 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | 2−1+4 | D7 |
| hypocenter | C7 | #577 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 258 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)(8,10,12,17)(9,14,13,20)(11,19,18,16)(15,23,21,22);(9,13)(14,20)(15,21)(22,23);(8,9)(10,14)(11,15)(12,13)(16,22)(17,20)(18,21)(19,23);(2,3)(4,5)(6,7)(10,16)(11,18)(14,22)(15,21)(17,19)(20,23)Polycyclic presentation:
2^6,7|d^2=f,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 | #11737732, #11737734, #11737746, … |
| maximal normal subgroup of | #11737732, #11737734, #11737746, … |
| maximal quotient of | #11739992, #11739993, #11739995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1290);