Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C7×#322 ⋊ C2 |
| Sylow | C7 ⋊ #1452 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D7 |
| commutator, socle | C22×C7 | C2×C42 |
| Fitting | C7×#322 | C2 |
| prime radical | C22×D7 | C42 |
| cosocle | C42×C7 | C23 |
| hypercenter | #322 | D7 |
| hypocenter | C7 | #1452 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C42 ◁ #322 | |
196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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=d,b^2=e,d^2=f,e^2=g,b^a=bg,c^b=cf,h^c=h^6176 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#1452, C2×#11725825, #13319402, … |
| maximal normal subgroup of | D7×#1452, C2×#11725825, #13319402, … |
| maximal quotient of | #13243160, #13243180, #13244224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 591);