Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C7 ⋊ #2709 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting, prime radical | C2×C7×#127 | C2 |
| hypercenter | C2×#127 | D7 |
| hypocenter | C7 | #2709 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
612 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 141 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|b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^6116 conjugacy classes — view the full character table.
| maximal subgroup of | #13272757, #13272759, #13273234, … |
| maximal normal subgroup of | #13272757, #13272759, #13273234, … |
| maximal quotient of | #13281339, #13281887, #13281952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 15394);