Finite Group DatabasebetaFinite Group Databasebeta| order | 17216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×D4×C269 ⋊ C2 |
| Sylow | C269 ⋊ #546 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | C22×D4×C269 | C2 |
| hypercenter | C22×D4 | D269 |
| hypocenter | C269 | #546 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 4304 |
Polycyclic presentation:
2^6,269|a^2=e,b^a=be,c^a=cf,g^a=g^268,d^b=de2705 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #46822358, #46822365, #46822395, … |
| maximal normal subgroup of | #46822358, #46822365, #46822395, … |
| maximal quotient of | D4×C22⋊Dic269, #46813710, #46813730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17216, 1193);