Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×Q32 ⋊ C2 |
| Sylow | C7 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7×D8 |
| commutator, cosocle | C7×C8 | C23 |
| Frattini | C8 | C22×D7 |
| Fitting | C7×Q32 | C2 |
| socle | C2×C7 | C2×D8 |
| hypercenter | Q32 | D7 |
| hypocenter | C7 | #508 |
| LCS | G ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
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)(8,10,13,18)(9,15,14,29)(11,23,26,39)(12,28,24,21)(16,34,30,38)(17,35,27,32)(19,37,36,22)(20,33,31,25);(8,9,13,14)(10,19,18,36)(11,22,26,37)(12,27,24,17)(15,31,29,20)(16,33,30,25)(21,38,28,34)(23,35,39,32);(2,3)(4,5)(6,7)(9,14)(10,18)(16,30)(17,27)(20,31)(21,28)(22,37)(23,39)Polycyclic presentation:
2^6,7|b^2=f,c^2=f,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 | #11727133, #11727134, #11727137, … |
| maximal normal subgroup of | #11727133, #11727134, #11727137, … |
| maximal quotient of | #11726960, #11726967, #11726968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 453);