Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×B(4,𝔽2) ⋊ C2 |
| Sylow | C7 ⋊ #3026 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7×#120 |
| commutator, cosocle | C23×C7 | C24 |
| Frattini | C23 | C23×D7 |
| Fitting | C7×B(4,𝔽2) | C2 |
| socle | C2×C7 | C2×#120 |
| hypercenter | B(4,𝔽2) | D7 |
| hypocenter | C7 | #3026 |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
800 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^7,7|b^a=be,c^a=cf,d^a=dg,f^b=fg,e^c=eg,h^d=h^677 conjugacy classes — view the full character table.
| maximal subgroup of | #13507850, #13507852, #13507861, … |
| maximal normal subgroup of | #13507850, #13507852, #13507861, … |
| maximal quotient of | #13327627, #13327634, #13327641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 12518);