Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C83 ⋊ C2 |
| Sylow | C83 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D83 |
| commutator, socle | C22×C83 | C24 |
| Fitting | C22⋊Q8×C83 | C2 |
| hypercenter | C22⋊Q8 | D83 |
| hypocenter | C83 | #554 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,83|a^2=e,b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de,g^d=g^82599 conjugacy classes — view the full character table.
| maximal subgroup of | #554×D83, C2×#19128405, #24921015, … |
| maximal normal subgroup of | #554×D83, C2×#19128405, #24921015, … |
| maximal quotient of | #24910335, #24910337, #24910361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1253);