Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C7 ⋊ #2978 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D7 |
| commutator, socle | C22×C7 | C23×C4 |
| Fitting, prime radical | C7×#525 | C2 |
| cosocle | C2×C4×C7 | C24 |
| hypercenter | #525 | D7 |
| hypocenter | C7 | #2978 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C4 ◁ #525 | |
402 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 197 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|a^2=f,f^2=g,b^a=be,c^a=cg,d^a=de,h^a=h^6,c^b=ceg,d^b=dg122 conjugacy classes — view the full character table.
| maximal subgroup of | #13250745, #13250749, #13250752, … |
| maximal normal subgroup of | #13250745, #13250749, #13250752, … |
| maximal quotient of | #13352502, #13352523, #13352581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 15132);