Finite Group DatabasebetaFinite Group Databasebeta| order | 17216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C269 ⋊ #556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | #124×C269 | C2 |
| prime radical | C2×D4×C269 | C22 |
| hypercenter | #124 | D269 |
| hypocenter | C269 | #556 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #124 | |
212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 8608 |
Polycyclic presentation:
2^6,269|b^2=ef,c^2=e,d^2=e,b^a=be,c^a=cf,d^b=de,g^c=g^268,g^d=g^2681901 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #46823529, #46823531, #46823565, … |
| maximal normal subgroup of | #46823529, #46823531, #46823565, … |
| maximal quotient of | #46813775, #46813833, #46813837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17216, 1268);