Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | D4×GL(3,𝔽2) |
| unambiguous semidirect product | C4⋊(C2×GL(3,𝔽2)) |
| unambiguous semidirect product | C22⋊(C2×GL(3,𝔽2)) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C22×GL(3,𝔽2) |
| commutator, socle | C2×GL(3,𝔽2) | C22 |
| Fitting, hypercenter | D4 | GL(3,𝔽2) |
| hypocenter | GL(3,𝔽2) | D4 |
| LCS | G ▷ C2×GL(3,𝔽2) ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 ◁ D4 | |
251 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 28 |
30 conjugacy classes — view the full character table.
| maximal subgroup of | S4×GL(3,𝔽2), SD16×GL(3,𝔽2), #72244484, … |
| maximal normal subgroup of | SD16×GL(3,𝔽2), #72244484 |
| maximal quotient of | SD16×GL(3,𝔽2), #72221408, D4×SL(2,𝔽7), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11301);