Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C19×C22⋊Q8 ⋊ C2 |
| Sylow | C19 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D19 |
| commutator, socle, cosocle | C22×C19 | C24 |
| Fitting | C19×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D19 |
| hypocenter | C19 | #548 |
| LCS | G ▷ C22×C19 ▷ C19 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
290 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 608 |
Polycyclic presentation:
2^6,19|b^2=e,d^2=e,b^a=be,c^a=cf,d^b=de,g^b=g^18,g^c=g^18151 conjugacy classes — view the full character table.
| maximal subgroup of | #14575128, #14575137, #14575187, … |
| maximal normal subgroup of | #14575128, #14575137, #14575187, … |
| maximal quotient of | #14566469, #14566472, #14566554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 1209);