Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C7 ⋊ #436 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D7 |
| commutator, socle | C22×C7 | C22×C4 |
| Fitting | C4×C7×D4 | C2 |
| prime radical | C2×C7×D4 | C4 |
| cosocle | C2×C4×C7 | C23 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #436 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(13,16)(17,19);(8,9)(10,11)(13,16)(17,19);(2,3)(4,5)(6,7)(12,13,14,17,15,16,18,19)Polycyclic presentation:
2^6,7|a^2=d,d^2=f,b^a=bf,c^a=cf,g^a=g^6,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #11725907, #11725909, #11725912, … |
| maximal normal subgroup of | #11725907, #11725909, #11725912, … |
| maximal quotient of | #11727202, #11727204, #11727205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 546);