Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C7×H(3,𝔽4) ⋊ C2 |
| Sylow | C7 ⋊ #1655 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6516 |
| commutator, cosocle | C42×C7 | C23 |
| Frattini | C42 | C22×D7 |
| Fitting | C7×H(3,𝔽4) | C2 |
| socle | C22×C7 | #120 |
| hypercenter | H(3,𝔽4) | D7 |
| hypocenter | C7 | #1655 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ H(3,𝔽4) | |
262 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^7,7|d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^6,d^b=df,e^b=eg,d^c=dg,e^c=efg77 conjugacy classes — view the full character table.
| maximal subgroup of | #13274525, #13274526, D7×#1655, … |
| maximal normal subgroup of | #13274525, #13274526, D7×#1655, … |
| maximal quotient of | #13242494, #13242541, #13242554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4468);