Finite Group DatabasebetaFinite Group Databasebeta| order | 1216 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C19×#120 ⋊ C2 |
| Sylow | C19 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10597383 |
| commutator, cosocle | C23×C19 | C23 |
| Frattini | C23 | C22×D19 |
| Fitting | C19×#120 | C2 |
| socle | C2×C19 | #120 |
| hypercenter | #120 | D19 |
| hypocenter | C19 | B(4,𝔽2) |
| LCS | G ▷ C23×C19 ▷ C2×C19 ▷ C19 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 152 |
Polycyclic presentation:
2^6,19|b^a=bd,c^a=ce,e^b=ef,d^c=df,g^c=g^18142 conjugacy classes — view the full character table.
| maximal subgroup of | #14565619, #14565620, #14565625, … |
| maximal normal subgroup of | #14565619, #14565620, #14565625, … |
| maximal quotient of | #14562551, #14562552, #14562556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1216, 735);