Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×#474 ⋊ C2 |
| Sylow | C7 ⋊ #3354 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D7 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#474 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #474 | D7 |
| hypocenter | C7 | #3354 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #474 | |
386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|a^2=e,b^2=g,c^2=eg,f^2=g,b^a=beg,c^a=cf,d^a=dg,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 | #13344696, #13344698, #13344741, … |
| maximal normal subgroup of | #13344696, #13344698, #13344741, … |
| maximal quotient of | #13370222, #13370253, #13371580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 13202);