Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×D16 ⋊ C2 |
| Sylow | C7 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7×D8 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×D16 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | D16 | D7 |
| hypocenter | C7 | #508 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D16 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,4,6,7,5,3)(9,15)(10,19)(11,23)(12,24)(14,27)(16,33)(18,32)(20,21)(22,38)(25,39)(26,37)(28,35)(29,36)(30,34);(8,10)(9,14)(11,19)(12,26)(13,22)(15,18)(16,32)(17,35)(20,23)(21,24)(25,38)(27,30)(28,34)(29,31)(33,36)(37,39);(2,3)(4,5)(6,7)(8,9)(10,18)(11,16)(12,17)(13,15)(14,22)(19,27)(20,36)(21,35)(23,34)(24,31)(25,30)(26,29)(28,37)(32,38)(33,39)Polycyclic presentation:
2^6,7|d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^6,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef55 conjugacy classes — view the full character table.
| maximal subgroup of | #11727130, #11727131, #11727133, … |
| maximal normal subgroup of | #11727130, #11727131, #11727133, … |
| maximal quotient of | #11726929, #11726934, #11726938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 446);