Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#108 ⋊ C2 |
| Sylow | C7 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6458 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#108 | C2 |
| socle | C2×C7 | #121 |
| hypercenter | #108 | D7 |
| hypocenter | C7 | #472 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #108 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,11,13,17,14,16,12,9)(10,18,20,23,21,15,19,22);(8,10,13,20,14,21,12,19)(9,15,16,23,17,18,11,22);(2,3)(4,5)(6,7)(8,9)(10,15)(11,12)(13,16)(14,17)(18,21)(19,22)(20,23)Polycyclic presentation:
2^6,7|b^2=e,c^2=df,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^6,c^b=ce,e^c=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | #11726996, #11726997, #11727012, … |
| maximal normal subgroup of | #11726996, #11726997, #11727012, … |
| maximal quotient of | #11726292, #11726294, #11726297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 431);