Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | (C2×#3866)⋊Q8 |
| Fitting | C2×Q8×C37 ⋊ C4 |
| Sylow | C37 ⋊ #391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#3866 |
| commutator, socle | C22×C37 | C22×C4 |
| Fitting | C2×Q8×C37 | C4 |
| prime radical, cosocle | C22×D37 | C23 |
| hypercenter | C2×Q8 | #3866 |
| hypocenter | C37 | #391 |
| LCS | G ▷ C22×C37 ▷ C37 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
186 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 592 |
Polycyclic presentation:
2^6,37|a^2=f,b^2=def,c^2=e,b^a=bd,c^a=ce,g^a=g^6,g^b=g^31,g^f=g^36118 conjugacy classes — view the full character table.
| maximal subgroup of | #18009692, #18010443, #18010552, … |
| maximal normal subgroup of | #18009692, #18010443, #18010552, … |
| maximal quotient of | #18009773, #18009775, #18009781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 1502);