Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C23×C3×C13 ⋊ C22 |
| Hall | C3×C13 ⋊ H(5,𝔽2) |
| Sylow | C13 ⋊ #1004 |
| Sylow | C3 ⋊ #88251 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D13 |
| commutator, socle, cosocle | C2×C3×C13 | C24 |
| Fitting | C23×C3×C13 | C22 |
| hypercenter | C23 | D3×D13 |
| hypocenter | C3×C13 | H(5,𝔽2) |
| LCS | G ▷ C2×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C2 ◁ C23 | |
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 312 |
Polycyclic presentation:
2^5,3,13|d^a=de,g^a=g^12,c^b=ce,f^b=f^2135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1357);