Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| isoclinism | isoclinic class of #89261 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ #347 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×Dic7 |
| commutator | C4×C7 | C42 |
| Frattini | C2×C8 | C2×D7 |
| Fitting | C7×M32 | C2 |
| socle | C2×C7 | #97 |
| prime radical | C22×C7 | M16 |
| cosocle | C2×C7×C8 | C22 |
| hypercenter | M32 | D7 |
| hypocenter | C7 | #347 |
| LCS | G ▷ C4×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ M32 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,4,6,7,5,3)(8,9,10,15,11,13,17,20,12,14,16,21,18,19,23,22);(2,3)(4,5)(6,7)(9,13,14,19)(10,16)(15,22,21,20)(17,23)Polycyclic presentation:
2^6,7|a^2=c,b^2=d,d^2=e,e^2=f,b^a=be,d^a=df,g^a=g^6,c^b=cf82 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#347, #11726841, #11726868, … |
| maximal normal subgroup of | D7×#347, #11726841, #11726868, … |
| maximal quotient of | #11725317, #11725318, #11725617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 69);