Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#135 ⋊ C2 |
| Sylow | C7 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#1141 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×#135 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | #135 | D7 |
| hypocenter | C7 | #509 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
90 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)(8,9)(10,17)(11,20)(12,14)(13,16)(15,18)(19,22)(21,23);(9,14)(13,21)(15,20)(17,22);(2,3)(4,5)(6,7)(9,13)(10,16)(11,18)(14,21)(15,22)(17,20)(19,23)Polycyclic presentation:
2^6,7|d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^6,c^b=cf,d^c=de,e^c=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | #11726996, #11726997, #11727003, … |
| maximal normal subgroup of | #11726996, #11726997, #11727003, … |
| maximal quotient of | #11728403, #11728406, #11728407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 727);