Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C19×#137 ⋊ C2 |
| Sylow | C19 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D19 |
| commutator, cosocle | C4×C19 | C24 |
| Frattini | C4 | C23×D19 |
| Fitting | C19×#137 | C2 |
| socle | C2×C19 | C22×D4 |
| hypercenter | #137 | D19 |
| hypocenter | C19 | #577 |
| LCS | G ▷ C4×C19 ▷ C2×C19 ▷ C19 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #137 | |
258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 304 |
Polycyclic presentation:
2^6,19|a^2=f,e^2=f,b^a=be,e^a=ef,e^b=ef,g^b=g^18,d^c=df,g^d=g^18121 conjugacy classes — view the full character table.
| maximal subgroup of | #14575400, #14575404, #14575405, … |
| maximal normal subgroup of | #14575400, #14575404, #14575405, … |
| maximal quotient of | #14572159, #14572167, #14572229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 1358);