Finite Group DatabasebetaFinite Group Databasebeta| order | 17216 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×D4×C269 ⋊ C2 |
| Sylow | C269 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D269 |
| commutator, socle, cosocle | C23×C269 | C23 |
| Fitting | C22×D4×C269 | C2 |
| hypercenter | C22×D4 | D269 |
| hypocenter | C269 | #392 |
| LCS | G ▷ C23×C269 ▷ C269 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
322 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 69 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^a=bd,c^a=ce,c^b=cf,g^c=g^2682702 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #46809801, #46809806, #46809828, … |
| maximal normal subgroup of | #46809801, #46809806, #46809828, … |
| maximal quotient of | #46809124, #46809126, #46809127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17216, 457);