Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#97 ⋊ C2 |
| Sylow | C7 ⋊ #432 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C7×#97 | C2 |
| prime radical | C22×D7 | C2×C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | #97 | D7 |
| hypocenter | C7 | #432 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ #97 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(9,13)(12,15)(16,17,18,21,19,20,22,23);(8,10,11,14)(9,12,13,15)(17,20)(21,23);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^6,7|b^2=e,c^2=d,d^2=f,c^a=ce,g^a=g^6,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #11726097, #11726098, #11726099, … |
| maximal normal subgroup of | #11726097, #11726098, #11726099, … |
| maximal quotient of | #11726359, #11726361, #11726410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 239);