Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C7×D8 ⋊ C2 |
| Sylow | C7 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×D8 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | C2×D8 | D7 |
| hypocenter | C7 | #576 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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)(10,16)(11,19)(14,22)(15,21)(17,18)(20,23);(8,10)(9,14)(11,16)(12,18)(13,20)(15,22)(17,19)(21,23);(9,13)(14,20)(15,21)(22,23);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,13)(16,22)(17,23)(18,20)(19,21)Polycyclic presentation:
2^6,7|e^2=f,b^a=bf,g^a=g^6,d^c=de,e^c=ef,e^d=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #11727012, #11727013, #11729225, … |
| maximal normal subgroup of | #11727012, #11727013, #11729225, … |
| maximal quotient of | #11736171, #11736176, #11736182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1210);