Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C83 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D83 |
| commutator, socle | C22×C83 | C24 |
| Fitting | C4×D4×C83 | C2 |
| hypercenter | C4×D4 | D83 |
| hypocenter | C83 | #554 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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^b=g^82845 conjugacy classes — view the full character table.
| maximal subgroup of | #24906533, #24906537, #24906595, … |
| maximal normal subgroup of | #24906533, #24906537, #24906595, … |
| maximal quotient of | #24910338, D4×C4⋊Dic83, #24910368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 1256);