Finite Group DatabasebetaFinite Group Databasebeta| order | 5312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C83 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D83 |
| commutator, socle | C23×C83 | C23 |
| Fitting | #95×C83 | C2 |
| hypercenter | #95 | D83 |
| hypocenter | C83 | #400 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ #95 | |
134 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 55 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=d,b^2=de,c^2=d,b^a=bd,c^a=ce,g^a=g^82,c^b=cf842 conjugacy classes — view the full character table.
| maximal subgroup of | #400×D83, C2×#19127642, #24910721, … |
| maximal normal subgroup of | #400×D83, C2×#19127642, #24910721, … |
| maximal quotient of | #24909023, #24909028, #24909033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5312, 490);