Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#423 ⋊ C2 |
| Sylow | C7 ⋊ #3036 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D28 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#423 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #423 | D7 |
| hypocenter | C7 | #3036 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #423 | |
548 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 145 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=e,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^a=h^6,f^c=fg,h^c=h^6116 conjugacy classes — view the full character table.
| maximal subgroup of | #13249730, #13249732, #13249744, … |
| maximal normal subgroup of | #13249730, #13249732, #13249744, … |
| maximal quotient of | #13337576, #13337587, #13337595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 11266);