Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #135×C37 ⋊ C2 |
| Sylow | C37 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D37 |
| commutator, cosocle | C4×C37 | C24 |
| Frattini | C4 | C23×D37 |
| Fitting | #135×C37 | C2 |
| socle | C2×C37 | C22×D4 |
| hypercenter | #135 | D37 |
| hypocenter | C37 | #578 |
| LCS | G ▷ C4×C37 ▷ C2×C37 ▷ C37 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 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) = 1184 |
Polycyclic presentation:
2^6,37|a^2=f,b^2=f,e^2=f,b^a=be,e^a=ef,e^b=ef,d^c=df,g^c=g^36,g^d=g^36274 conjugacy classes — view the full character table.
| maximal subgroup of | #17996557, #17996561, #17996571, … |
| maximal normal subgroup of | #17996557, #17996561, #17996571, … |
| maximal quotient of | #18003125, #18003193, #18003432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 1363);