Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C7×D8 ⋊ C2 |
| Sylow | C7 ⋊ #449 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D7 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C2×C7×D8 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×D8 | D7 |
| hypocenter | C7 | #449 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
162 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 224 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,6,7,5,3)(9,11)(13,17)(14,19)(16,18);(12,13)(14,17)(15,16)(18,19);(2,3)(4,5)(6,7)(8,9)(10,11)(13,16)(17,18)Polycyclic presentation:
2^6,7|e^2=f,b^a=bf,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 | #11737036, #11737038, #11737040, … |
| maximal normal subgroup of | #11737036, #11737038, #11737040, … |
| maximal quotient of | #11727918, #11727923, #11727953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 692);