Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C4×C7×C8 ⋊ C2 |
| Sylow | C7 ⋊ #433 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C4×C7×C8 | C2 |
| prime radical | C22×D7 | C2×C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C4×C8 | D7 |
| hypocenter | C7 | #433 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,4,6,7,5,3)(9,13)(12,15)(16,18,19,24,20,25,26,30)(17,22,23,29,21,27,28,31);(8,10,11,14)(9,12,13,15)(17,21)(22,27)(23,28)(29,31);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,22)(19,23)(20,21)(24,29)(25,27)(26,28)(30,31)Polycyclic presentation:
2^6,7|b^2=e,c^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^6124 conjugacy classes — view the full character table.
| maximal subgroup of | #11735893, C2×#89462, #11735900, … |
| maximal normal subgroup of | #11735893, C2×#89462, #11735900, … |
| maximal quotient of | #11726360, #11726362, #11726410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 224);