Finite Group DatabasebetaFinite Group Databasebeta| order | 1248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #88251⋊C3 |
| Fitting | C2×D4×C13 ⋊ C2×C3 |
| Hall | #88251 ⋊ C3 |
| Hall | C13⋊C3 ⋊ H(5,𝔽2) |
| Sylow | C13 ⋊ C3×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D13⋊C3 |
| commutator, socle, cosocle | C2×C13 | C24×C3 |
| Fitting | C2×D4×C13 | C2×C3 |
| hypercenter | C2×D4 | D13⋊C3 |
| hypocenter | C13 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
332 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^4,3,2,13|d^a=df,g^a=g^12,c^b=cf,g^e=g^371 conjugacy classes — view the full character table.
| maximal subgroup of | H(5,𝔽2)×D13⋊C3, #72229068, #72259180, … |
| maximal normal subgroup of | H(5,𝔽2)×D13⋊C3, #72229068, #72259180, … |
| maximal quotient of | #72259622, #72259629, #72259641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1248, 1248);