Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C7×D8 ⋊ C2 |
| Sylow | C7 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6501 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C2×C7×D8 | C2 |
| socle | C2×C7 | #121 |
| hypercenter | C2×D8 | D7 |
| hypocenter | C7 | #472 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(9,15)(10,18)(11,12)(14,21)(16,17)(19,20);(8,10,12,19,13,20,11,18)(9,14,16,22,17,21,15,23);(2,3)(4,5)(6,7)(8,9)(10,14)(11,15)(12,16)(13,17)(18,23)(19,22)(20,21)Polycyclic presentation:
2^6,7|b^2=e,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^6,c^b=ce,d^c=df,e^c=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | #11727014, #11727015, #11729225, … |
| maximal normal subgroup of | #11727014, #11727015, #11729225, … |
| maximal quotient of | #11727949, #11727950, #11727951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 694);