Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#102 ⋊ C2 |
| Sylow | C7 ⋊ #447 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D7 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#102 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | #102 | D7 |
| hypocenter | C7 | #447 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
198 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(8,9,10,11)(13,16)(14,19)(17,18);(12,13)(14,16)(15,18)(17,19);(2,3)(4,5)(6,7)(9,11)Polycyclic presentation:
2^6,7|c^2=d,e^2=f,c^a=cd,g^a=g^6,c^b=ce,e^b=ef,e^c=ef61 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#89545, #11736192, #11736202, … |
| maximal normal subgroup of | C2×#89545, #11736192, #11736202, … |
| maximal quotient of | #11725934, #11725936, #11725938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 307);