Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#131 ⋊ C2 |
| Sylow | C7 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6493 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#131 | C2 |
| socle | C2×C7 | #121 |
| hypercenter | #131 | D7 |
| hypocenter | C7 | #472 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 | |
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)(8,10)(9,15)(11,18)(12,19)(13,20)(14,17)(16,23)(21,22);(8,9,14,21,12,16,11,13)(10,15,17,22,19,23,18,20);(2,3)(4,5)(6,7)(9,13)(10,17)(11,14)(16,21)(18,19)(20,22)Polycyclic presentation:
2^6,7|b^2=ef,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^6,d^c=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #11726880, #11726881, #11728846, … |
| maximal normal subgroup of | #11726880, #11726881, #11728846, … |
| maximal quotient of | #11727646, #11727648, #11727657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 676);