Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C19 ⋊ #389 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D19 |
| commutator | C22×C19 | C22×C4 |
| Fitting | C2×C42×C19 | C2 |
| socle, prime radical, cosocle | C23×C19 | C23 |
| hypercenter | C2×C42 | D19 |
| hypocenter | C19 | #389 |
| LCS | G ▷ C22×C19 ▷ C19 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
170 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1216 |
Polycyclic presentation:
2^6,19|a^2=f,b^2=f,c^2=e,b^a=bd,c^a=ce,g^a=g^18316 conjugacy classes — view the full character table.
| maximal subgroup of | #14561457, #14561459, #14563882, … |
| maximal normal subgroup of | #14561457, #14561459, #14563882, … |
| maximal quotient of | #14561865, #14561868, #14561887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 447);