Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×M32 ⋊ C2 |
| Sylow | C7 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D56 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×M32 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | M32 | D7 |
| hypocenter | C7 | #509 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32 | |
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)(8,9,15,21,11,16,18,19,12,14,22,23,17,20,10,13);(9,14)(13,19)(16,20)(21,23);(2,3)(4,5)(6,7)(9,13)(10,15)(11,17)(14,19)(16,23)(18,22)(20,21)Polycyclic presentation:
2^6,7|c^2=de,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^6,c^b=cf76 conjugacy classes — view the full character table.
| maximal subgroup of | #11726845, #11726874, #11726880, … |
| maximal normal subgroup of | #11726845, #11726874, #11726880, … |
| maximal quotient of | #11726823, #11726831, #11726832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 442);