Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #5429 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10621482 |
| commutator, cosocle | C7×#41⋊C3 | C22 |
| Frattini, hypercenter | C4 | D7×S4 |
| Fitting | C7×#41 | C2×D3 |
| socle | C2×C7 | GL(2,ℤ/4) |
| hypocenter | C7×SL(2,𝔽3) | D4 |
| LCS | G ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
156 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^3,3,7,2^3|a^2=h,c^2=h,f^2=h,g^2=h,b^a=bc,c^a=ch,e^a=e^6,c^b=ch,d^b=d^2,f^b=fg,g^b=gh,f^d=g,g^d=fgh,g^f=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#5429, #72246270 |
| maximal normal subgroup of | D7×#5429, #72246270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6381);