Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, prime radical, cosocle | C22×C7 | C24 |
| Fitting | C4×C7×Q8 | C2 |
| hypercenter | C4×Q8 | D7 |
| hypocenter | C7 | #557 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,11,15)(9,12,13,14);(16,19,20,17)(18,21,22,23);(16,18,20,22)(17,21,19,23);(2,3)(4,5)(6,7)(8,9,11,13)(10,14,15,12)(16,17,20,19)(18,21,22,23)Polycyclic presentation:
2^6,7|a^2=e,b^2=f,c^2=f,d^2=ef,d^a=def,g^a=g^6,c^b=cf85 conjugacy classes — view the full character table.
| maximal subgroup of | #11726006, #11726007, #11726010, … |
| maximal normal subgroup of | #11726006, #11726007, #11726010, … |
| maximal quotient of | #11731085, #11731096, #11731105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1023);