Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#135 ⋊ C2 |
| Sylow | C7 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×#135 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | #135 | D7 |
| hypocenter | C7 | #576 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
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)(9,13)(10,17)(11,20)(14,22)(16,23)(18,19);(8,10)(9,14)(11,17)(12,19)(13,16)(15,23)(18,20)(21,22);(8,9)(10,16)(11,15)(12,13)(14,19)(17,22)(18,23)(20,21);(2,3)(4,5)(6,7)(9,13)(14,16)(15,21)(22,23)Polycyclic presentation:
2^6,7|e^2=f,b^a=bf,g^a=g^6,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #11726996, #11726998, #11729271, … |
| maximal normal subgroup of | #11726996, #11726998, #11729271, … |
| maximal quotient of | #11736265, #11736268, #11736272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1222);