Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C19×#135 ⋊ C2 |
| Sylow | C19 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D19 |
| commutator, cosocle | C4×C19 | C24 |
| Frattini | C4 | C23×D19 |
| Fitting | C19×#135 | C2 |
| socle | C2×C19 | C22×D4 |
| hypercenter | #135 | D19 |
| hypocenter | C19 | #576 |
| LCS | G ▷ C4×C19 ▷ C2×C19 ▷ C19 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 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|e^2=f,b^a=be,e^a=ef,e^b=ef,d^c=df,g^c=g^18,g^d=g^18148 conjugacy classes — view the full character table.
| maximal subgroup of | #14565589, #14565593, #14565603, … |
| maximal normal subgroup of | #14565589, #14565593, #14565603, … |
| maximal quotient of | #14572157, #14572232, #14572462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 1351);