Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #90632⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #90632⋊C3 |
| Fitting, socle, Sylow | C26 ⋊ ΓL(1,𝔽8) |
| derived, Hall | #90632 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #90632 | C3 |
| Fitting, socle | C26 | ΓL(1,𝔽8) |
| LCS | G ▷ #90632 | |
| UCS | C1 | |
202 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 14 |
Polycyclic presentation:
3,7,2^6|b^a=b^2,c^a=cd,d^a=c,f^a=fg,g^a=f,c^b=cde,d^b=cd,e^b=de,f^b=gh,g^b=f,h^b=g16 conjugacy classes — view the full character table.
| maximal subgroup of | #72209654, #72209657, #72209659, … |
| maximal normal subgroup of | C2×#90632⋊C3, C3×#90632⋊C3 |
| maximal quotient of | #72209829, C2×#90632⋊C3, #72210723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11691);