Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#126 ⋊ C2 |
| Sylow | C7 ⋊ #563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#126 | C2 |
| prime radical | C2×C22⋊D7 | C22 |
| hypercenter | #126 | D7 |
| hypocenter | C7 | #563 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #126 | |
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,4,6,7,5,3)(8,10,11,14)(9,13,12,15)(17,23)(18,25)(19,27)(28,30);(9,12)(13,15)(16,19)(17,22)(18,24)(20,27)(21,28)(23,26)(25,31)(29,30);(8,9)(10,13)(11,12)(14,15)(16,18,20,25)(17,21,23,29)(19,24,27,31)(22,28,26,30);(2,3)(4,5)(6,7)(10,14)(13,15)(16,17)(18,21)(19,26)(20,23)(22,27)(24,30)(25,29)(28,31)Polycyclic presentation:
2^6,7|b^2=f,d^2=e,c^a=cf,d^a=def,g^a=g^6,c^b=ce,d^b=df,d^c=df61 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#90403, #11743178, #11743186, … |
| maximal normal subgroup of | C2×#90403, #11743178, #11743186, … |
| maximal quotient of | #11731710, #11731725, #11731726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1165);