Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#463 ⋊ C2 |
| Sylow | C7 ⋊ #3321 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#463 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #463 | D7 |
| hypocenter | C7 | #3321 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #463 | |
392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|a^2=e,b^2=g,c^2=g,d^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^6,c^b=cg,d^b=deg,h^b=h^6,f^c=fg,h^c=h^6,h^d=h^686 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#3321, C2×#11738009, #14230554, … |
| maximal normal subgroup of | D7×#3321, C2×#11738009, #14230554, … |
| maximal quotient of | #13369388, #13370096, #13370877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 12775);