Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C7×#127 ⋊ C2 |
| Sylow | C7 ⋊ #453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6500 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#127 | C2 |
| socle | C2×C7 | #120 |
| hypercenter | #127 | D7 |
| hypocenter | C7 | #453 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #127 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(9,12)(10,15)(13,14);(10,15)(12,13);(2,3)(4,5)(6,7)(8,9)(10,12)(11,14)(13,15)Polycyclic presentation:
2^6,7|d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^6,c^b=ce,d^b=df,d^c=df58 conjugacy classes — view the full character table.
| maximal subgroup of | #11725268, #11725344, #11726900, … |
| maximal normal subgroup of | #11725268, #11725344, #11726900, … |
| maximal quotient of | #11727327, #11727346, #11727347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 611);