Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| Fitting | C7×H(3,𝔽4) ⋊ C2 |
| Sylow | C7 ⋊ #3549 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C7×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D7 |
| hypocenter | C7 | #3549 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^7,7|b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^6,h^e=h^695 conjugacy classes — view the full character table.
| maximal subgroup of | #13335607, #13335611, #13335641, … |
| maximal normal subgroup of | #13335607, #13335611, #13335641, … |
| maximal quotient of | #13516646, #13517312, #13517318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 18921);