Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle, prime radical | C22×C7 | C22×C4 |
| Fitting | C4×C7×Q8 | C2 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C4×Q8 | D7 |
| hypocenter | C7 | #446 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,7,5,3)(8,10,11,15)(9,12,13,14)(17,20)(21,23);(8,9,11,13)(10,14,15,12);(2,3)(4,5)(6,7)(16,17,18,21,19,20,22,23)Polycyclic presentation:
2^6,7|a^2=d,b^2=e,c^2=e,d^2=f,c^a=cf,g^a=g^6,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #11725970, #11725973, #11725974, … |
| maximal normal subgroup of | #11725970, #11725973, #11725974, … |
| maximal quotient of | #11727231, #11727232, #11727236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 558);