Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×C7×#123 ⋊ C2 |
| Sylow | C7 ⋊ #2697 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×#123 | C2 |
| prime radical | C2×#6500 | C2 |
| hypercenter | C2×#123 | D7 |
| hypocenter | C7 | #2697 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 137 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=e,b^2=e,b^a=bef,c^a=cf,d^a=de,c^b=cg,d^b=def,h^c=h^6116 conjugacy classes — view the full character table.
| maximal subgroup of | #13272738, #13272740, #13273215, … |
| maximal normal subgroup of | #13272738, #13272740, #13273215, … |
| maximal quotient of | #13281084, #13281195, #13281326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 15119);